The Oxford Companion to Philosophy Part 59 potx

mechanisms that determine social change. For other philosophers it represents a science of society, based on a well-grounded ontology (human activities and social con- sciousness are the basis of existence) and a well-defended ‘realist’ epistemology (knowledge is derived empirically from social interactions with the material world). More radically, post-modern Marxists have reconceptualized Marxism as grounded in a democratic and egalitarian civil society. This cuts Marxism loose from historical teleology and philosophical determinism, while retaining its critical engagement with politics. t.car. R. Bhaskar, Scientific Realism and Human Emancipation (London, 1986). G. A. Cohen, Karl Marx’s Theory of History: A Defence, expanded. edn. (Oxford, 2000). E. Laclau and C. Mouffe, Hegemony and Socialist Strategy: Towards a Radical Democratic Politics, 2nd edn. (London, 2001). L. Sargent (ed.), Women and Revolution: A Discussion of the Unhappy Marriage of Marxism and Feminism (London, 1981). Marxist philosophy. The idea of a Marxist philosophy is, at first sight, paradoxical. Marx himself was originally a student of philosophy but soon came to talk of abolishing philosophy: the coming of a socialist society would render philosophy (like religion) redundant. It is, nevertheless, clear that Marx and his followers appropriated much of the philosophy of (at least) Aristotle, the *materialism of the *Enlightenment, and Hegelian dialectics. It is equally clear that when Marx talked of the abolition of philoso- phy, he meant that, in so far as philosophy posed ideal principles or essences, it would lose its function after a socialist revolution which embodied these essences in socio-economic reality. It is far from clear that Marx’s *his- torical materialism contradicts or supersedes philosophy as such. The century and more that has elapsed since Marx’s death has been a largely fruitless search by his fol- lowers to establish a distinctively Marxist philosophy. Since the authoritarian Communist regimes established in Marx’s name did not encourage philosophical enterprise, their demise is unlikely to have much effect on the future of Marxist philosophy. Although Marx himself had apparently disparaged phil- osophy, after his death and with the revolution still a long way off, the ‘footnotes to Plato’ had to be dealt with and the growing membership of Marxist parties required a ‘philosophy’ in the sense of a coherent system of principles giving a total explanation of the universe. Given the cul- tural climate of the late nineteenth century, this had to be couched in scientific—and even positivist—terms. Although the later Marx certainly had traces of such atti- tudes in his work, it was given systematic form by Engels and culminated in the philosophy of dialectical material- ism propagated by Communist orthodoxy. Engels proclaimed the Marxian *dialectic to be ‘the sci- ence of the general laws of motion and development of nature, human society, and thought’. More specifically, the most important of these were the laws of the transform- ation of quantity into quality, of the interpretation of opposites, and of the negation of the negation. Engels thought these laws to be operative in a nature that was objectively given and independent of the human mind. Thus the world of nature and the world of human history were two separate fields of study—whereas for Marx one of the central aspects of his dialectic had consisted pre- cisely in the interaction of human beings and their sur- roundings, a view stemming from Hegel. Engels did indeed claim to be simply applying Hegel’s dialectic, and, in a sense, Hegel also saw a dialectic in nature but it was still subject to the universal mediation of human con- sciousness. The concept of matter as some kind of materia prima is entirely foreign to Marx. For many interpreters of Marx’s thought, however, the publication of his early writings around 1930 marked a decisive turning-point. These writings, particularly the Economic and Philosophical Manuscripts, revealed a very dif- ferent Marx from both the rather arid economist of Kaut- sky and the dialectical materialist of Soviet dogma. Marx appeared to be a philosopher, a humanist not only with a devastating account of the alienation of man in capitalist society but with a rich and varied account of the potential latent in every individual waiting to be realized under communism. This enthusiasm for the early Marx was helped by the pioneering writings of Georg Lukács, who rediscovered in full Marx’s debt to Hegel and put concepts such as alienation and reification at the centre of his inter- pretation. This tradition has been embodied most system- atically in the work of the *Frankfurt School, where ‘critical’ theorists such as Adorno, Marcuse, and Haber- mas have aimed to restore a philosophical dimension to Marxism. Retaining an enviable confidence in the power of human rationality, these theorists have developed a series of concepts intended to go beyond Marx in inter- preting the changes that have taken place in the world since his death. These consist mainly in adding the dimen- sion of social psychology to Marx’s work, and emphasiz- ing the basic proposition that if society is increasingly under the control of technocrats, then any purely empiri- cal approach to social reality must end up as a defence of that control. In sharp contrast to the evidently Hegelian and human- ist elements present in the Marxism of the Frankfurt School, the Marxist philosophy evolved by Althusser and his disciples in the early 1960s attempted to purge Marx- ism of any such elements. Taking advantage of the current prestige of structuralist linguistics, psychology, and anthropology, it was the aim of Althusser to ‘rehabilitate’ Marx as a structuralist before his time. Thus Althusser continued the Stalinist division of an early pre-Marxist Marx and a later scientific Marx—though with a concep- tual sophistication quite foreign to the previous versions of this view. Roughly speaking, *structuralism is the view that the key to the understanding of a social system is the structural relationship of its parts—the way these parts are related by the regulative principle of the system. And Althusser’s search for a timeless rationality reminiscent of Comte (for whom Marx himself had no time) involved the 560 Marxism after communism banishment of both history and philosophy. When applied to Marx, this involved cutting his work into two separate conceptual structures with the dividing-point around 1845. Any reading of Marx as a humanist, a Hegelian, or a historicist must (since these ideas are clearly contained in his early works) be rejected. Since it has become increasingly implausible to claim (particularly after the publication of the Grundrisse) that there are no humanist or Hegelian elements in the later Marx, a ‘real’ Marx has been uncovered who employs a methodology— never clearly defined—almost totally at variance with concepts that he actually employs. More recently, there have been attempts to rethink many aspects of Marxism through the medium of rational choice theory. This approach, exemplified in the writings of such authors as Elster and Roehmer, has come to be known as analytical Marxism. Central to it is a *method- ological individualism which borrows concepts and tech- niques from game theory and contemporary economics. Particularly when combined with *analytical philosophy, this can yield a highly rigorous discussion. But the concep- tual framework is so much at variance with the Marxist tradition that it is not surprising that the theses of the ana- lytical Marxist school are highly revisionary. And the same is true of Marxist attempts to come to terms with the rise of new social movements, particularly those inspired with an ecological or feminist perspective. The most striking fact about the relation between Marxism and philosophy, in the West at least, is how eclectic Marxists have been in their attitude to philosophy. Marxists have usually tried to articulate their ideas through whatever happened to be the current dominant philosophy. The revival of interest in Hegel between the wars, coupled with the influence of Freud, was decisive for the formulations of the Frankfurt School; the post-war vogue for existentialism led to all sorts of New Left vari- ations on Marxism with a human face, of which Sartre’s later work is only the most prominent example; the subse- quent prestige of structuralism in the 1960s and 1970s led to the arcanely theoretical Marxism of Althusser and his disciples; while the rational choice Marxism of more recent years is evidently an effort to come to terms with some of the dominant concepts of the Reagan–Thatcher years. The inevitable tension in all the above approaches lies in the fact that all the philosophies they invoke are the product of bourgeois societies—the very societies that Marxism is dedicated to superseding. This tension is only exacerbated by the tendency of western Marxists to become more theoretical and more philosophical with the decreasing prospect of success for Marxist practical activ- ity. The migration of Marxism into the universities has necessarily undercut the unity of theory and practice so central to the outlook of Marx himself. For him, all philosophy (like all religion) is ultimately idealist and mystificatory. Holding that ‘the dispute over the reality or non-reality of thinking that is isolated from practice is a purely scholastic question’, Marx looked forward to a society which would abolish the division between mental and manual work—which he saw as the root cause of all philosophical mystification. Such a society would be intel- ligible to its members, since the social relationships in it would be transparent, and would not require philosoph- ical mediation. The history of Marxist thought has thus been charac- terized by a strong ambivalence towards the viability of the philosophical enterprise. The result has been the invisibility of a distinctively Marxist philosophy: Marxism has been eclectic in its borrowings from ‘bourgeois’ phil- osophy. These borrowings have been extremely fruitful, particularly in the realm of social theory. Indeed here, as elsewhere, Marxism has proved at its strongest as a cri- tique of philosophy rather than in adumbrating a plausible alternative. d.m cl. *socialism; communism; anti-communism. P. Anderson, Considerations on Western Marxism (London, 1976). R. Bhaskar, A Realist Theory of Science (Brighton, 1978). K. Korsch, Marxism and Philosophy (London, 1970). J. Mepham and D H. Ruben (eds.), Issues in Marxist Philosophy (Brighton, 1979). New Left Review, Western Marxism: A Critical Reader (London, 1977). Mary, black and white. Omni-competent in the neuro- physiology of vision, Mary is confined to a black-and- white environment. However omniscient she becomes in the neurophysics of the experience of ripe tomatoes, the blue sky, etc., she will find, if released into the chromatic world, that she still has something to discover: namely, what the relevant visual experiences are like. The thought experiment was designed to show that physicalism is false, and discussion has turned, among other things, on whether the kind of knowledge Mary lacks is propos- itional at all, let alone knowledge of a non-physical fact. a.h. Frank Jackson, ‘Epiphenomenal Qualia’, Philosophical Quarterly (1982). Masaryk, Tomásˇ Garrigue (1850–1937). Czech philoso- pher, sociologist, and politician who influenced gener- ations of Czech and Slovak thinkers. From 1882 to 1914 he was Professor at Charles University in Prague, from 1918 to 1935 President of the fledgeling Czechoslovakia. He was an opponent of clericalism, monarchism, anti- Semitism, and Bolshevism. He sought to explain the crisis in Czech and European society at the end of the nineteenth century, a significant feature of which was an increase in suicide (The Suicide . . . (1881)). In The Principles of Concrete Logic (1885) he follows Comte’s classification of sciences, adding logic and psy- chology, as understood in J. S. Mill’s terms. He also dealt with Czech history, and the struggle of every human being and society to attain the human ideal of active love. This religious humanity was more emotional than the political and rational humanity of the French Revolution. In The Social Question (1898) he rejects Marxism as Masaryk, Tomásˇ 561 objectivism, positivistic amoralism, and materialistic fatalism, although he admits its temporary political and ideological sense. In The Spirit of Russia (1913) he analyses Russian culture in the nineteenth century, emphasizing the thinking of Dostoevsky. m.p. Jan Patocˇka, Three Studies about Masaryk (Prague, 1991). masculism. Defining ‘masculism’ is made difficult by the fact that the term has been used by very few people, and by hardly any philosophers. In its most general meaning, the word *‘feminism’ refers to promotion of the interests or rights of women, and a reasonable definition of ‘mas- culism’ would have it refer to promoting the interests or rights of men. (This is very different, it must be noted, from promoting attributes of womanliness or manliness, as they might be construed, which could be labelled femi- ninism and masculinism.) Thus defined, the two parallel terms are too vague to be very useful. A more precise def- inition of both would be something on this order: ‘the belief that women/men have been systematically dis- criminated against, and that that discrimination should be eliminated’. Evidently, such a definition for ‘feminism’ is commonly understood, and among the few who apply the term ‘masculist’ to themselves, such is also their intent. Of course, under these meanings there is no necessary con- flict between them, and in fact some are happy to call themselves both feminists and masculists. Much more often, the belief that one sex currently faces a much greater threat from discrimination would lead to accept- ing one label and rejecting the other. However one understands these particular terms, there is today a small movement of ‘men’s rights’ activists. Their fundamental claim is that very serious discrimin- ation is currently being committed against individual males on account of their sex. These activists fall roughly into two categories, traditionalist and liberal–progressive. The traditionalists hold that inherited gender roles, though ‘discriminatory’ in the neutral sense of treating the sexes differently, have been more or less fair and just to both, because, they believe, the disadvantages faced by males and females have been comparable (at least in this culture, in this century) and because the traditional sex roles repre- sent more or less the optimal division of benefits and bur- dens, the best arrangement for children and for society as a whole. What sets ‘men’s rights’ traditionalists apart from traditionalists in general is their belief that contem- porary feminism is not only bad for society but seriously unjust to men as well. In sharp contrast—and in spite of attempts by many to label all talk of men’s rights as reactionary, a ‘backlash’— progressive men’s rights activists regard the traditional differential treatment as seriously unfair to members of both sexes. Inherited *gender roles and stereotypes are not just burdensome to both men and women, they say, but unjust to both, and must be eliminated. (Unlike trad- itionalists, they have no need to pronounce the roles equally burdensome, and tend to treat the two sets of injustices as incommensurable.) Progressive masculists have thus welcomed many feminist efforts toward soci- etal change, adding, however, that feminism addresses only half the problem. Furthermore, they maintain that many feminist efforts ostensibly aimed at ending sexism are actually increasing sexism against men. This has been especially true, they say, in the 1980s and 1990s, as main- stream feminism has left its inclusivist roots in favour of separatist efforts based on an extreme oppressor– oppressed picture of relationships between the sexes. Thus, both forms of contemporary masculism promote equality between men and women as its adherents envi- sion it. Of course, whether they are mistaken about what moral equality would consist in, or even at some level dis- honest about that being their goal, is another matter—as it also is for feminists. This leads us to the extremist versions of masculism and feminism, those that promote some degree of male or female supremacy, and are generally based on belief in the inferiority of the other sex. Many contemporary feminists consider men to be morally and even intellectually inferior, by virtue of being raised in an oppressor class, or even by nature. And of course the long history of male domination since hunter-gatherer times has generally included doctrines of the intellectual infer- iority, and, although the record is mixed, sometimes moral inferiority of women. Nicholas Davidson discusses an extreme brand of masculism and masculinism which he dubs ‘virism’. In its world-view, he says, What ails society is ‘effeminacy’. The improvement of society requires that the influence of female values be decreased and the influence of male values increased. . . . Contemporary virists per- ceive themselves to be fighting a last-ditch action against a neutered or feminized society, of which feminism is merely one recent expression. . . . [In movies such as] Rambo and Commando, the world has gone soft. The protagonists struggle to avert dan- gers caused by society’s loss of the masculine principle. Davidson sees precise similarities between extremist mas- culism and extremist feminism, remarking that ‘the paral- lel association of Hellenic virism with a cult of [male] homosexuality and modern feminism with a cult of lesbianism is not accidental’. However that may be, most men’s and women’s rights activists profess belief in *equality, different though their visions of it may be. Indeed, they do not divide strictly along gender lines. Besides feminist (or ‘pro-feminist’) men, there are many women—some embracing the label ‘feminist’, some rejecting it, and many ambivalent about it—who actively advocate men’s rights. Such groups as the Women’s Freedom Network (mostly libertarians) and the Women’s International Network (liberal) in the USA have been established largely to oppose the harms they see contemporary mainstream feminism as doing to both sexes. Traditionalist women’s groups such as Eagle Forum (USA) and REAL Women (Canada) also often speak out against discrimination toward men, or at least against the recent varieties promoted by feminists. Space is not available here to describe adequately (much less to argue for and against) the standard men’s rights issues. They include discrimination against fathers 562 Masaryk, Tomásˇ in child custody cases (in terms of numbers of activists, this is the largest issue); discrimination against men in the criminal law, military conscription, and various other societal institutions; contemporary discrimination against men in employment, insurance and pensions, and other economic matters; and many others. (See Farrell, The Myth of Male Power, and Thomas, Not Guilty, for represen- tative treatments of men’s rights issues.) The above discussion describes masculism as a set of political beliefs, not as philosophy in any abstract sense. Apart from advocacy (genuine, not just alleged) of male supremacy, however, there arguably is no masculist phil- osophy. Consider the traditionalist belief that if nature were allowed to take its course, men would fill most of the leaderships roles (see Goldberg, Why Men Rule) and women fill the nurturing roles. The belief is better described as a general philosophy of human nature than as one centred on males and maleness. And liberal advocates for men’s rights typically describe their philosophy as egali- tarian rather than as either male- or female-orientated. By the same reasoning, however, apart from brands of femi- nism embracing genetic female superiority, there is no genuine *feminist philosophy, or at least none with unique relevance to females or femaleness. The perfectly justified desire to open up to women the opportunities, which only men (a small minority of men) have had in the past, to engage in formal philosophy, had led, this writer would judge, to the wishful beliefs that (a) past philosophy, in virtue of having been written by individual males, is somehow specifically male or masculist in its nature, and that (b) there is a distinct type of philosophy that is specifi- cally female in its nature. All the post-Gilligan talk about women’s ‘special ways of seeing’ notwithstanding, as this liberal masculist–feminist writer views the evidence, *‘feminist epistemology’ and its like is a grand illusion. f.chr. Nicholas Davidson, The Failure of Feminism (Buffalo, NY, 1988). Warren Farrell, The Myth of Male Power (New York, 1993). Steven Goldberg, Why Men Rule (Chicago, 1993). David Thomas, Not Guilty: The Case in Defence of Men (London, 1993). masked man fallacy. One of a group of puzzles (the *liar paradox is another) due to Eubulides (third century bc), the masked man, extensively discussed by medieval logi- cians, is concerned with referentially opaque contexts: ‘You say you know your brother, but that masked man is your brother, and you did not know him’. By contrast, normal (extensional) contexts are transparent: if you touched the masked man you thereby touched your brother. j.j.m. W. V. Quine, Word and Object (Cambridge, Mass., 1960), ch. 4. master and slave. The master and slave metaphor has occurred in philosophy since ancient times. Aesop admonished that reason should be the master and passion the slave. (Hume was to say otherwise.) Politically, of course, the duality of master and slave and later lord and serf were quite literal social realities for Aristotle and most feudal thinkers. The master and slave imagery enters into modern philosophy in Rousseau, Fichte, and most famously in Hegel, who made the master–slave inter- action the centrepiece of the most famous section of his Phenomenology. According to Hegel, the master–slave rela- tionship is the result of an uncompleted fight to the death for ‘recognition’ or status, and it is marked by a topsy- turvy logic (or *‘dialectic’) such that the master, through increasing dependence on the slave, and the slave, who develops independence through labour, switch roles. Hegel is at pains to point out that such roles, dependent on competition and power, are unsatisfactory as a basis for social relationships. Toward the end of the nineteenth century, Nietzsche picked up the opposition as a metaphor for two distinct modes of morality. Master morality prizes independence, creativity, and excellence. *Slave morality, by contrast, is servile, fearful, and, above all, resentful. In a Hegelian- type turn-about, Nietzsche traces ethics through history as ‘the slave revolt in morals’. Slave morality is victorious, and, according to Nietzsche, servile obedience and medi- ocrity replace the masterly Greek ideals of virtue and excellence. r.c.sol. G. W. F. Hegel, The Phenomenology of Spirit (Oxford, 1977). F. Nietzsche, On the Genealogy of Morals (New York, 1967). Master Argument. The Master Argument was highly influential on Hellenistic debates about *freedom and determinism. Diodorus Cronus (Greek, d. c.284 bc) devised it to support his definition of the possible as that which either is or will be true. Diodorus relied on two pre- misses: ‘Every past truth is necessary’ and ‘The impossible does not follow from the possible’. He concluded: ‘Noth- ing is possible which neither is nor will be true’. One guess, less fanciful than most, is that Diodorus reasoned: suppose some proposition is and always will be untrue; then at some time in the past, it was going to be untrue at all later times; from that past truth, it follows that our proposition is untrue; but that past truth is necessary; and so is what follows from it; hence our proposition is necessarily untrue. n.c.d. *fatalism. Nicholas Denyer, ‘Time and Modality in Diodorus Cronus’, Theoria (1981). master of those who know: see Aristotelianism. material constitution. That which something is com- posed of. For example, the statue is not identical with the bronze it is made of, but the bronze is the material consti- tution of the statue. Nevertheless, there is a constitutive sense of ‘is’. Arguably, something is not identical with what it is made of because the two have different persist- ence conditions. If a is composed of constituents, they might continue to exist even if a ceases to exist. Also, if a is composed of constituents, and if a ceases to exist, and then material constitution 563 b is made out of those same constituents, it seems wrong to say a is therefore b. On some quasi-materialist views, if I am not identical with my body, my body is my material constitution. s.p. John Locke, An Essay Concerning Human Understanding, book ii. D. Wiggins, Sameness and Substance (Oxford, 1980). material contradiction. The idea that *contradictions exist not only in thought but in material reality has been a distinctively philosophical preoccupation within *Marx- ism. It refers to activities within organisms and systems that generate opposing forces (thus, ‘capitalism creates its own grave-diggers’), and also the claim, adapted from Hegel, that adequate descriptions of material reality necessarily involve contradictions. It need not, therefore, imply a metaphysical realist conception of contradictions as having an extra-linguistic existence. k.m. R. Norman and S. Sayers, Hegel, Marx and Dialectic: A Debate (Brighton, 1980). material implication. A connection between statements which has sometimes seemed to be supposed by logicians to represent statements of the form ‘If . . . then . . . ’. A state- ment p materially implies another, q, when the propos- itional calculus conditional p → q is true, i.e. if and only if it is not the case that p is true and q false. One ‘paradox’ of material implication is that this relation holds between statements wholly unrelated in subject-matter: ‘If Oxford is a city, then Italy is sunny’. Another ‘paradox’ is that the relation holds merely if p is false (‘If pigs can fly, then . . . ’) or merely if q is true (‘If . . . then Plato was a philosopher’). These are all ways in which material implication diverges from ‘if . . . then . . . ’ as ordinarily used. s.w. *Implication; conditionals; relevance logic P. F. Strawson, Introduction to Logical Theory (London, 1952), chs. 2 and 3. materialism. Basically the view that everything is made of matter. But what is *matter? Probably the most innocent and cheerful acceptance of it comes right at the start of materialism with Democritus of Abdera (in northern Greece) in the fifth century bc, for whom the world con- sisted entirely of ‘atoms’, tiny, absolutely hard, impene- trable, incompressible, indivisible, and unalterable bits of ‘stuff’, which had shape and size but no other properties and scurried around in the void, forming the world as we know it by jostling each other and either rebounding (despite being incompressible) or getting entangled with each other because of their shapes. They and the void alone were real, the colours and flavours and tempera- tures that surround us being merely subjective (see fragment 9). This model has lasted, with various modifica- tions and sophistications, right down until modern times, though the notion of solidity was causing qualms at least as early as Locke. But in the last century all has been thrown into confusion by Einstein’s famous E = mc 2 and also by general relativity. Mass, the sophisticated notion that has replaced crude matter, is interchangeable in cer- tain circumstances with *energy, and in any case is only a sort of distortion of the *space in which it was supposed to be floating. Photons and neutrons have little or no mass, and neither do fields, while particles pop out of the void, destroy each other, and pop back in again. All this, however, has had remarkably little overt effect on the various philosophical views that can be dubbed ‘materialism’, though one might think it shows at least that materialism is not the simple no-nonsense, tough- minded alternative it might once have seemed to be. What actually seems to have happened is that the various materialist philosophies have tended to substitute for ‘matter’ some notion like ‘whatever it is that can be studied by the methods of natural science’, thus turning materialism into *naturalism, though it would be an exaggeration to say the two outlooks have simply coin- cided. Materialism concerns the composition of things, while naturalism, though concerned with what exists, ranges more widely, covering properties as well as sub- stances, and its concern with methods of studying things is more direct and central. So far, however, we have only considered apparent exceptions to naïve materialism like photons and fields; but common sense might regard these as on the material side of the fence, since they are linked to what is unam- biguously material in established and fruitful scientific theories. In any case they are of little interest to common sense. Philosophers too, outside philosophy of science perhaps, often take as their starting-point the concerns of common sense, and perhaps it is for these reasons that materialism in philosophy has also been relatively unaffected by these complications. By far the most important contrast in this area, for phil- osophy as for common-sense, has been between matter and mind or spirit or consciousness, or the contents of these entities (ideas etc.). Common-sense for all the time and philosophers for much of it (i.e. when they are not being idealists or phenomenalists) accept the reality of the body as relatively unproblematic. But mind or conscious- ness manifestly exists too in some form, at least in our own case. Does it then exist as a separate entity? This question forms a major part of the *mind–body problem, and the materialist will answer ‘No’. He might hard-headedly deny that mind exists at all, thereby raising a question that has its roots in Protagoras of Abdera (a contemporary of Democritus) and Descartes: what then is the status of the illusion that it does? Don’t even illusions need minds to have them? But he is more likely to appeal to some form of non-eliminative materialism and say that minds do exist but not as something separate from matter. Either they belong to a different category, as Ryle would say, so that talk about them is a sort of abbreviation for talk about kinds of behaviour (*behaviourism), or they are simply identical with brains—or, to put it less crudely, what we call mental phenomena, like pains or thoughts, are identi- cal with phenomena going on in the brain (the *identity theory of mind): in principle this can leave open whether 564 material constitution there are any substantive minds to have these pains etc., but if there are they must be identical with something material. Although the material item will be identical with the mind or the pain as much as vice versa, the theory counts as materialist because the material items are inte- grated into a whole set of such items, only a few of which are involved with minds, while the mental items are not (on the theory) similarly integrated into a set of mental items most of which are independent of matter. There is, however, a mistake to be avoided in this area which goes back at least to Plato. In his relatively early dia- logue the Phaedo he opposes the soul (to adopt the usual translation of the Greek word psukhe¯) and the body. The soul is a single thing (unlike the tripartite soul of his slightly later dialogue the Republic), but it is represented as in conflict with the body, and pulled by bodily desires and passions. These desires and passions are opposed to the soul as such, and are clearly regarded as bodily phenom- ena, which they are surely not, at least on the sort of view Plato is holding; human bodies no more have desires than tables do. It is true, of course, that the identity theory we have just been discussing would treat these desires as iden- tical with events in the brain or nervous system, but only as part of a doctrine treating all desires as such in this way, including the most spiritual; it would not single out cer- tain desires for such treatment merely because what the desires were for involved states of the body, or what caused them were such states. In his later dialogue the Philebus Plato revised his view on this point. Sometimes a distinction is made, as it has been in differ- ent ways by Frege and Popper, between three kinds of real things (three ‘realms’ or ‘worlds’). The first contains mater- ial things, including the things like photons that are asso- ciated with centrally material things and might be called quasi-material. The second contains psychological things like thoughts, feelings, pains, desires, including the sub- stantive minds that have these, if there are any; if there are, then the thoughts etc. that they have will not be independ- ent substantive entities, but will still count as denizens of the second realm. The third contains abstract things like numbers, properties, classes, truths (and perhaps false- hoods), values, or some selection of these, all those selected being treated as substantive entities, though not material nor properly speaking spiritual either. Philoso- phers have tended to treat these three realms not so much as lying in a straight line, with one in the middle between the other two, as lying in a triangle, so that the rejection of one was compatible with accepting either or both of the other two. Materialists strictly speaking say that only mat- ter exists, but in modern times they have tended to direct their fire primarily against believers in the second realm, and some of them (e.g. Armstrong) accept at least a mod- erate realism in connection with the third realm. But this was not always so. Plato, with whom so much of phil- osophy began, was primarily concerned to assert the existence of the third realm (indeed he is commonly taken to have introduced it, and belief in it is often called *Platonism), and in his dialogue the Sophist he contrasts materialists with certain defenders of that realm. Although he devoted another dialogue, the Phaedo, to defending the immortality of the soul, a member of the second realm, its existence he tended to take for granted, leaving its status vis-à-vis the other two realms rather uncertain. However, despite all this it is perhaps true that most materialists are thoroughgoing and reject both the other two realms, even if directing their fire primarily at one of them. We started our discussion by asking ‘What is matter?’ and finding that our notion of matter was getting a bit frayed at the edges. There is a further difficulty with it, which appears to be what led Berkeley to his version of *idealism, which he called ‘immaterialism’. Aristotle con- strasted matter with form, and treated it as a *substratum for form and thus ultimately for attributes. This led him, at least as traditionally interpreted, to a notion of ‘prime mat- ter’, which was the ultimate subject for all attributes and therefore had no attributes of its own. Locke, again as traditionally interpreted, took over this notion of prime matter and made it the underlying but unknowable *substance of all things—unknowable because it had no attributes by which we could know it. Actually Locke’s position on this question is open to dispute, but Berkeley rejected the notion as simply ridiculous and a source of scepticism because of the bewilderment it led to. Evi- dently the materialist, and indeed everyone who accepts matter at all, must give some account of the nature of mat- ter which will rescue it from these strictures. That this task will not be easy is suggested by the growing revival of idealism in current philosophy. So far we have considered materialism as a meta- physical doctrine. But in ordinary thinking it often refers to a doctrine about values. Here again it is often con- trasted with idealism, which now refers to the pursuit of ideals that may be high-flown but are likely to be impos- sible to achieve in practice. The materialist by contrast pursues ends connected with the bodily pleasures, or the possession of material goods, or else with such things as money, thought of as a means to such pleasures or goods. One might ask, however, just what count as bodily pleas- ures. If they are those involving states of the body, what happens to the aesthetic pleasures of music and visual art? We should be hard put to it to enjoy these without ears or eyes, and these are not being used simply to give us inform- ation as when we hear or read poetry (or philosophy for that matter); to enjoy a piece of music is not simply to know what it sounds like, but to hear it sounding like that, even if only in the ‘mind’s ear’. It is also possible that the term ‘materialist’ involves some confusion due to the sort of considerations we discussed concerning Plato above. But in any case we should not confuse two distinctions: that between pleasures that involve the body more closely and those that involve it less closely or not at all, and that between values that are in some sense ‘lower’ and less worthy of pursuit and those that are ‘higher’. The contrast between materialism and idealism in this context suggests such a confusion, but the pleasures, or values, of the materialism 565 materialist are not necessarily any ‘lower’, presumably, than those of the pursuit of, say, malice. a.r.l. *atomism; physicalism; phenomenalism; Quinton; central-state materialism; behaviourism; elimin- ativism. S. E. Toulmin and J. Goodfield, The Architecture of Matter (London, 1962). Historical. G. Amaldi, The Nature of Matter (first pub. in Italian, 1961; London, 1966). Scientific. D. M. Armstrong, Universals: An Opinionated Introduction (Boulder, Colo., 1989). A. Melnyk, A Physicalist Manifesto (Cambridge, 2003). materialism, central-state: see central-state materialism. materialism, dialectical: see dialectical materialism. materialism, eliminative: see eliminativism. materialism, historical: see historical materialism. material mode: see formal and material modes. mathematical intuitionism: see intuitionism, math- ematical. mathematical logic: see formal logic. mathematics, history of the philosophy of. Many areas of philosophy owe their beginning to Plato, and the phil- osophy of mathematics is a star example. It was he who first reflected on the fact that the geometers speak of perfect squares, perfect circles, and so on, though no examples are to be found in this world. He thought that the same applied to arithmetic too, on the ground that in arithmetic we study numbers that are composed of units perfectly equal to one another in every way, whereas again there are no such units to be found in this world. So he con- cluded that mathematics was not about the objects to be found in this world, but about some different ‘purely intel- ligible’ objects, which he was apt to think of as inhabiting ‘another world’. Moreover, since the objects were not of this world, our knowledge of them must also be independ- ent of our experience of this world, i.e. it must be *a priori. Initially he attempted to explain this a priori knowledge as a recollection of our past experiences of the other world, before birth. Later he appears to have aban- doned this explanation, but he never ceased to insist that the knowledge is a priori. Mathematics, then, has both a special ontology and a special epistemology. Aristotle challenged both of these claims. In his philos- ophy there was no place for a distinctive kind of know- ledge that could be called a priori, and he equally could not accept Plato’s ontological extravagance. So he was the first to propose a consciously ‘reductive’ account of the objects of mathematics. In his view the geometer is speak- ing of ordinary and perceptible squares and circles, but considered generally, and in abstraction from the ordinary and perceptible matter that they are made of. Similarly, the arithmetical thesis that 2 + 3 = 5 is to be construed simply as a generalization over such ordinary facts as that if there are 2 horses in this field, and 3 in that, then there are 5 horses altogether. But unfortunately he never did set out a full argument for these claims, so the rival positions were outlined, but battle was never properly joined. Space does not permit an account of the skirmishes between what were broadly speaking Platonic and Aris- totelian positions in the centuries that followed, and we may pick up the tale again with the quite different outlook of the ‘modern’ period. There we find a dispute between *rationalism on the one side and *empiricism on the other, but it is a dispute in which there is actually a large measure of agreement. The ontology is shared, for on both sides the objects of mathematics are taken to be our ideas. And the epistemology is at least partly shared. The difference is that the rationalists suppose (as Plato once did) that the relevant ideas are innate, whereas the empiri- cists think that our idea of three, or of a triangle, owes its existence to our perceptions of three-membered groups and triangular objects. But both are agreed that, once the relevant ideas are obtained, the further pursuit of math- ematical knowledge is independent of any further experi- ence. Yet despite all this agreement there is nevertheless an important opposition: rationalists such as Descartes stress the importance of mathematics for our understand- ing of the world, whereas Empiricists such as Locke and Berkeley and Hume belittle it. Descartes first extends the sphere of mathematics, so that it includes time, and hence motion, as well as space. Then, just as he supposes that the basic principles of (Euclidean) geometry are known a priori, so he presumes that the same will apply to the laws of motion, and he offers an a priori derivation of them. (Descartes’s sup- posed laws use only spatio-temporal concepts, such as size and velocity.) With this as a basis he professes to be able to deduce, without any aid from observation, the organiza- tion of the solar system as a whole (i.e. as a system of ‘vortices’). He also promises us that these same basic laws can in principle explain all further phenomena, from the behaviour of light to the action of the heart. The whole of science, then, in its completed form, will be just an appli- cation of a priori reasoning from innate principles. This is perhaps the boldest view that there has ever been of the scope and power of pure mathematics. Of course, parts of Descartes’s system were soon found to be wrong, and it needed Newton to produce a much better system, which apparently did square with observation. Moreover, Newton himself did not suppose that either his laws of motion or his law of gravitation could claim any a priori status. Instead, he made a point of citing experiment and observation in their support. But, even after Newton’s ‘corrections’, Descartes’s vision of a wholly a priori science remained a temptation to many (including Kant). It did not tempt Locke, or Berkeley, or Hume. Locke, indeed, was drawn to think that science proper should be a priori (for only so could it provide genuine explanations), but for that very reason he thought that we should never 566 materialism attain it. For all three Empiricists were well aware that in practice science must be based upon observation and experiment (and they all failed to see the importance of theorizing). But Berkeley and Hume actually attacked mathematics itself, claiming that it was based upon assumptions about infinite divisibility which had no basis in experience, and which in fact led to intolerable para- doxes. In so far as these charges concerned the notion of the *infinitesimal, there was indeed some truth in them, and they were not properly answered until Weierstrass (see below). But in any case, Hume’s general attitude should be noted: our knowledge of mathematics (so far as it is genuine) is knowledge of ‘relations between ideas’, and this is to be contrasted with knowledge of ‘matters of fact and existence’. Hume was interested in the latter. But Kant returned to the former, asking what kinds of ‘relations between ideas’ these were, and how they were discovered. Kant thought of an *analytic truth as one which can be established simply by an analysis of the concepts involved. Since he supposed that conceptual analysis was the mental analogue of taking something to bits, he framed his criter- ion in terms of concepts and their parts: an analytic truth is one in which the concept of the predicate is contained within the concept of the subject. His point was that it is not surprising that such truths can be known a priori. But he also supposed that some synthetic truths can be known a priori, his chief example being the truths of math- ematics, and his main problem was to explain how this could be so. He seems to have thought that, once we are clear about the distinction between the analytic and the synthetic, we will readily agree that the truths of geometry and arithmetic are indeed synthetic. At any rate, his argu- ments on this point are very superficial. But with hindsight we can now say that he was certainly right about geom- etry. When this is understood, with Kant, as a theory about the space in which we find ourselves, then certainly its basic assumptions are not analytic. But the difficulty here is with the other claim, that they are known a priori. Kant’s argument is just that we cannot imagine things otherwise, and that is why we feel that the knowledge is independent of experience, and could not be falsified by experience. He adds a ‘justification’ for this last point, namely that the spatial arrangement of what we perceive is our contribution to the interpretation of the data of sense. So it is due to our own nature, and not to the nature of the data, that we cannot perceive in any other way. This account of geometry was forcibly challenged by J. S. Mill. He admitted that we could not imagine things otherwise, but explained that this was due simply to the weakness of our imaginations, which were in practice limit- ed by what we had experienced and how we had under- stood it. To back this up, he pointed to several instances, from the history of science, of other propositions which once could not be imagined otherwise, but were later rejected (e.g. Aristotle’s laws of motion). And he inferred from this that what can be imagined may change as scientific theories change, and is no safe guide to necessary truth. His application of this line of thought to geometry is nicely done, and we can now say that on this topic he was definitely right. The mathematical development of *non- Euclidean geometries has provided strong support for his view that if Euclidean geometry does have any special status then that can only be because of its fit with experi- ence; and the fact that modern physical theory actually prefers a non-Euclidean geometry clearly vindicates his refusal to draw any conclusion from the limits to what can be imagined. Mill’s views on geometry are now orthodox. Both Kant and Mill attempted to treat arithmetic in the same way as they had treated geometry, Kant claiming that it was both synthetic and a priori, Mill that it was (syn- thetic and) empirical. But in this case the arguments on both sides were quite unconvincing, and were well criti- cized by Frege (see below). The second half of the nineteenth century saw two important developments in mathematics which together provoked a surge of philosophical thinking amongst the mathematicians themselves, and gave rise to three philosophies of mathematics which are influential even today. The one development began with what is known as ‘the arithmetization of analysis’. First Weierstrass (1815–97), building upon work by Canchy (1789–1857), succeeded in reducing to good order the differential and integral calculus introduced long ago by Newton and Leibniz. For two centuries it had been clear that this method led to many extremely useful results, but its basis had remained mysterious, apparently relying on the incomprehensible notion of the infinitesimal. Weierstrass showed how this notion could simply be eliminated. This work on the ‘foundations’ of a branch of mathematics was soon carried further. Both Dedekind and Cantor offered foundations for the theory of real numbers, by which it was freed from reliance on geometrical intuition, and derived instead from the theory of rational numbers. (Some *set theory was also employed in the derivation; this point received little attention at the time.) The theory of rational *numbers in turn could easily be derived from the theory of the natural numbers, and Dedekind went on to produce a foundation for this latter, i.e. a proper axio- matization of elementary arithmetic. So apparently Dedekind had now provided a foundation for ‘all’ of trad- itional mathematics. But the second development was in a different direction: Cantor created an entirely new branch of mathematics, i.e. the theory of *infinite numbers, which certainly could not be derived from these same foundations. Yet it cried out for foundations of its own, for it appeared to spawn contradictions (e.g. Cantor’s para- dox) no better than those that had once characterized the differential calculus. So on the one hand there was, for the first time since Greek mathematics, a new emphasis on foundational thinking; and on the other there was a new branch of mathematics, standing outside the area covered by existing foundations, and evidently controversial. This led to three new philosophies of mathematics, namely *logicism, a special variety of *formalism due to Hilbert, and *intuitionism. mathematics, history of the philosophy of 567 The founder of logicism was Frege. Though he was sympathetic to Cantor’s theory of the infinite, in his main work he concentrated just upon elementary arithmetic. In opposition to both Kant and Mill, he redefined an analytic truth as one that can be proved just from general logical laws and definitions, and set out to show that arithmetic is in this sense analytic. That is, it needs only logic for its foundation. At the same time he rejected the prevailing *‘psychologism’, i.e. the assumption that arithmetic is about our ideas, and reinstated the Platonic conception of numbers as abstract objects, existing quite independently of us. Putting these points together, he held that numbers were ‘logical objects’, and in effect he took them to be sets. This was the cause of his disaster, for it led him to adopt a very general (‘logical’) principle for the existence of sets, which Russell showed to be inconsistent. (*Russell’s para- dox.) Moreover, the inconsistency was clearly very similar to that affecting Cantor’s theory of the infinite, and so sub- sequent developments in the logicist tradition have gener- ally attempted to deal with both problems at once, providing a single foundation both for finite and for infi- nite numbers. One important example is Russell’s theory of *types, which at the same time abandons Frege’s Platonism about numbers. But in practice, mathematicians have preferred an explicit set theory as a foundation, and this strongly suggests the Platonist interpretation. In any case, logicism can in principle be combined with either the Platonic or the Aristotelian view of what numbers are. It is also compatible with either view on epistemology. For if we say that the main claim of logicism is that there is no firm boundary to be drawn between logic on the one side and mathematics on the other, then one can accept this claim while holding either that both have an a priori status (as Frege and Russell desired), or that both are empirical (as Quine once claimed). Hilbert is often counted as a formalist, but he differs from most others who are so-called because he wishes to apply the formalist approach, not to all mathematics, but only to some of it. He begins with the thought that Can- tor’s theory of the infinite should certainly be retained, but must be protected from contradiction in some way. At present it is vulnerable, because we do not really under- stand the notion of infinity, and so do not attach a clear ‘content’ to our reasoning. Then he generalizes this thought to all other areas of mathematics that involve infin- ite totalities, taking this to include not only the theory of the real numbers (since they may be construed as infinite sets, or infinite sequences, of rational numbers), but even the use of quantifiers in elementary arithmetic, since these quantifiers range over all the infinitely many natural num- bers. The most basic part of mathematics, then, is limited to what can be done in elementary arithmetic without quantifiers, using only free variables and recursive func- tions. This part has genuine content, is well understood, and can safely be assumed to be free from error. But in other areas we have no such guarantee. So his programme is first to formalize these other areas, proposing a formal theory that is adequate to represent all ordinary math- ematical reasoning in that area, and then to argue in the metalanguage that this formal theory is free from contra- diction. The metalinguistic argument, of course, must be confined to methods of proof that are guaranteed, i.e. to those already available in quantifier-free arithmetic. (This is one of several proposals on what may be counted as a ‘constructive’ method of proof. *Constructivism.) *Gödel’s incompleteness theorems destroyed this pro- gramme. The more far-reaching result was the first the- orem, which implies that, for any sufficiently rich formal system, there are methods of reasoning about that system which evidently do establish the truth of a formula of the system, but which go beyond what can be proved within the system. So one cannot even carry out Hilbert’s first step, of formalizing the area of mathematics to be proved consistent. But the second theorem shows that in any case the consistency of the system cannot be proved even by the methods available in that system itself, let alone by methods that Hilbert permitted as constructive. So Hilbert may perhaps be right in saying that, in some areas, all that the mathematician needs as a basis for his investi- gations is a consistent formal system, and that he need not attach any ‘content’ to that system. But we now know that we cannot use this idea to guarantee the safety of math- ematical reasoning. Brouwer, who founded intuitionism, shared Hilbert’s distrust of the infinite and his commitment to constructive reasoning. But whereas Hilbert had aimed to rescue the non-constructive parts of mathematics, Brouwer simply abandons them. Against formalism, then, he sees no merit in formal systems without true ‘content’, and against logi- cism he believes that mathematics is prior to logic, and does not need it. However, in 1930 his pupil Arend Heyt- ing did elaborate a logic suitable for intuitionistic reason- ing, which Brouwer then endorsed. It is quite different from classical logic, mainly because its leading idea is that truth cannot be distinguished from provability. (*Logic, intuitionist.) In Brouwer’s own thought one can find two bases for this idea. One is that mathematical objects have a special status: they are ‘mental constructions’. (This is a Kantian thought; it reverts to the ‘psychologism’ that Frege had attacked.) Since the objects do not exist inde- pendently of human thinking, it may seem to follow that we cannot adopt for them the same theory of truth as we use in other cases, namely a correspondence theory. In their case, then, there is nothing else for truth to be but provability. The other line of thought relies not on the alleged special status of the numbers, but just on the point that there are infinitely many of them. This means that, on the classical conception of truth, there could be some truth about all numbers which we could never verify, even in principle. But the intuitionist rejects this concep- tion of truth as no less ‘metaphysical’ than the Platonic conception of numbers as independently existing objects. Intuitionism is a revisionist theory in the philosophy of mathematics, for it involves abandoning much of classical mathematics. A more extreme form of this revisionism is ‘strict finitism’, which will not allow truth to go beyond 568 mathematics, history of the philosophy of what we can in practice verify. It is disputed whether Wittgenstein should be counted as a strict finitist. d.b. *formalism. P. Benacerraf and H. Putnam (eds.), Philosophy of Mathematics: Selected Readings, 2nd edn. (Cambridge, 1983). W. Ewald (ed.), From Kant to Hilbert: A Source Book in the Founda- tions of Mathematics (Oxford, 2000). M. Giaquinto, The Search for Certainity: A Philosophical Account of Foundations of Mathematics (Oxford, 2002). I. Grattan-Guinness (ed.), From the Calculus to Set Theory 1630–1910 (London, 1980). M. Kline, Mathematical Thought from Ancient to Modern Times (New York, 1972). W. and M. Kneale, The Development of Logic (Oxford, 1962). M. Potter, Reason’s Nearest Kin: Philosophies of Arithmetic from Kant to Carnap (Oxford, 2000). J. van Heijenoort (ed.), From Frege to Gödel (Cambridge, 1967). mathematics, knowledge of. Interpreted at face value, mathematics appears to be a science concerned with an extraordinary array of objects and structures that are abstract, necessary, and infinite (geometrical shapes and patterns, the natural numbers, the real numbers, the set- theoretic hierarchy, and so on). It is, however, deeply mysterious how knowledge of such objects and structures could ever be acquired by us, mundane and finite crea- tures that we appear to be. Of course, mathematicians employ proofs to confirm their assertions. But proofs only tell us what follow from the axioms of a mathematical the- ory; they cannot inform us that the axioms are true. A variety of different epistemological strategies have been proposed to overcome this difficulty. According to one popular strategy, mathematical axioms are confirmed by the indispensable (auxiliary) role they perform in scientific practice. But this ties the practice of mathematics to the empirical fortunes of science. To avoid this consequence, *logicism claims that mathematical axioms are truths of higher-order logic. s.s . *logicism; mathematics, problems of the philosophy of. Stewart Shapiro, Thinking about Mathematics (Oxford, 2000). mathematics, philosophy of, problems of. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to illu- minate the place of mathematics in our overall intellectual lives. The problems fall into categories familiar to con- temporary philosophers. There are ontological problems concerning the subject-matter of mathematics. What is mathematics about? There are epistemological problems. How do we know mathematics? What is the method- ology of mathematics, and to what extent is this method- ology reliable? There are problems of logic and semantics. How are the languages of mathematics understood, learned, communicated, etc.? What do mathematical assertions mean? Are unambiguous mathematical state- ments objectively true or false? What is the proper logic for mathematics? And there are problems concerning the relationship between mathematics and the rest of the intellectual enterprise. What is the relationship, if any, between the philosophy of mathematics and the practice of mathematics? (See the text by Shapiro and the collec- tion edited by Benacerraf and Putnam, listed in the selected bibliography.) Many of the problems and issues in mathematics are counterparts of central items on the agenda of general epis- temology and metaphysics. Mathematics provides a good case-study for many of the issues concerning existence, semantics, and knowledge that occur throughout philoso- phy. Nevertheless, mathematics at least appears to be dif- ferent in kind from other types of investigation. Its basic assertions enjoy an extremely high degree of certainty. Indeed, theorems of at least elementary mathematics, like ‘2 + 2 = 4’ or ‘There are infinitely many prime numbers’, are often taken to be paradigms of *necessary truths and *a priori and infallible knowledge. How can such things be false, and how can any rational being doubt them? Thus, it is incumbent on any complete philosophy of mathematics to account for the necessity and apriority of mathematics, or else to show why mathematics appears that way. Mathematics also plays an important role in virtually every scientific effort aimed at understanding the natural world. Consider, for example, the prerequisites of just about any natural or social science. It is thus incumbent on any complete philosophy of mathematics to show how mathematics is applied to the material world or, in other words, to show how the subject-matter of mathematics is related to the subject-matter of the sciences, and how the methodology of mathematics fits into the methodology of the sciences. There are two distinct types of *realism in the philoso- phy of mathematics. Realism-in-ontology is the view that the subject-matter of mathematics is a realm of objects that exist independent of the mind, conventions, and lan- guage of the mathematician. Most advocates of this view hold that mathematical objects—numbers, functions, points, sets, etc.—are abstract, eternal, and do not enter into causal relationships with material objects. Because of this, realism-in-ontology is sometimes called *‘Platon- ism’, noting the resemblance between mathematical objects and Platonic Forms. This label can be misleading, however. Realism-in-ontology does not, by itself, pre- suppose anything like a Platonic epistemology, and there are realists-in-ontology who hold that at least some math- ematical objects are not eternal and not outside the causal nexus. Realism-in-truth-value is the view that unambiguous assertions of mathematics are non-vacuously true or false, independent of the mind, language, and conventions of the mathematician. This view is sometimes called *realism. There is a natural connection between the two varieties of realism. Consider the following statement: There is a prime number greater than 1,000,000. The realist-in-truth-value holds that this is an objective truth. But what does it mean? Prima facie, ‘1,000,000’ is a mathematics, philosophy of, problems of 569 . applied to the material world or, in other words, to show how the subject-matter of mathematics is related to the subject-matter of the sciences, and how the methodology of mathematics fits into the. of the philosophy of. Stewart Shapiro, Thinking about Mathematics (Oxford, 2000). mathematics, philosophy of, problems of. The aim of the philosophy of mathematics is to provide an account of the. logic. mathematics, history of the philosophy of. Many areas of philosophy owe their beginning to Plato, and the phil- osophy of mathematics is a star example. It was he who first reflected on the fact that the geometers