Nominalism
Encyclopedia
Nominalism is a metaphysical
view in philosophy
according to which general or abstract terms and predicates
exist, while universals
or abstract object
s, which are sometimes thought to correspond to these terms, do not exist. Thus, there are at least two main versions of nominalism. One version denies the existence of universals—things that can be instantiated or exemplified by many particular things (e.g. strength, humanity). The other version specifically denies the existence of abstract objects—objects that do not exist in space and time.
Most nominalists have held that only physical particulars in space and time are real, and that universals exist only post res, that is, subsequent to particular things. However, some versions of nominalism hold that some particulars are abstract entities (e.g. numbers), while others are concrete entities—entities that do exist in space and time (e.g. tables, chairs).
Nominalism is primarily a position on the problem of universals
, which dates back at least to Plato
, and is opposed to realism—the view that universals do exist over and above particulars. However, the name "nominalism" emerged out of debates in medieval philosophy with Roscellinus
. As John Stuart Mill
once wrote, the early versions of nominalism were that "there is nothing general except names", hence the prefix "nomin-". This, however, is a more dated use of the term that is now considered to be a specific version of what is now called "nominalism".
was perhaps the first writer in Western philosophy to clearly state a non-Nominalist position, which he plainly endorsed:
The Platonic universals corresponding to the names "bed" and "beautiful" were the Form
of the Bed and the Form of the Beautiful, or the Bed Itself and the Beautiful Itself. Platonic Forms were the first universals posited as such in philosophy.
Our term "universal" is due to the English translation of Aristotle
's technical term katholou which he coined specially for the purpose of discussing the problem of universals. Katholou is a contraction of the phrase kata holou, meaning "on the whole".
Aristotle famously rejected certain aspects of Plato's Theory of Forms, but he clearly rejected Nominalism as well:
, specifically accounting for the fact that some things are of the same type. For example, Fluffy and Kitzler are both cats, or, the fact that certain properties are repeatable, such as: the grass, the shirt
, and Kermit the Frog
are green. One wants to know in virtue of what are Fluffy and Kitzler both cats, and what makes the grass, the shirt, and Kermit green.
The realist
answer is that all the green things are green in virtue of the existence
of a universal; a single abstract
thing, in this case, that is a part of all the green things. With respect to the color of the grass, the shirt and Kermit, one of their parts is identical. In this respect, the three parts are literally one. Greenness is repeatable because there is one thing that manifests
itself wherever there are green things.
Nominalism denies the existence of universals. The motivation for this flows from several concerns, the first one being where they might exist. Plato
famously held, on one interpretation, that there is a realm of abstract forms or universals apart from the physical world (see theory of the forms). Particular physical objects merely exemplify or instantiate the universal. But this raises the question: Where is this universal realm? One possibility is that it is outside of space and time. A view sympathetic with this possibility holds that, precisely because some form is immanent in several physical objects, it must also transcend each of those physical objects; in this way, the forms are "transcendant" only insofar as they are "immanent" in many physical objects. In other words, immanence implies transcendence; they are not opposed to one another. (Nor, on this view, would there be a separate "world" or "realm" of forms that is distinct from the physical world, thus shirking much of the worry about where to locate a "universal realm".) However, naturalists
assert that nothing is outside of space and time. Some Neoplatonists, such as the pagan philosopher Plotinus
and the Christian philosopher St. Augustine
, imply (anticipating conceptualism
) that universals are contained within the mind of God
. To complicate things, what is the nature of the instantiation
or exemplification
relation?
Conceptualists
hold a position intermediate between nominalism and realism
, saying that universals exist only within the mind
and have no external or substantial reality.
Moderate realists
hold that there is no realm in which universals exist, but rather universals are located in space and time wherever they are manifest. Now, recall that a universal, like greenness, is supposed to be a single thing. Nominalists consider it unusual that there could be a single thing that exists in multiple places simultaneously. The realist maintains that all the instances of greenness are held together by the exemplification relation, but this relation cannot be explained.
Finally, many philosophers prefer simpler ontologies
populated with only the bare minimum of types of entities, or as W. V. Quine said "They have a taste for 'desert landscapes.'" They attempt to express everything that they want to explain without using universals such as "catness" or "chairness."
Resemblance nominalists believe that 'cat' applies to both cats because Fluffy and Kitzler resemble an exemplar
cat closely enough to be classed together with it as members of its kind
, or that they differ from each other (and other cats) quite less than they differ from other things, and this warrants classing them together. Some resemblance nominalists will concede that the resemblance relation is itself a universal, but is the only universal necessary. This betrays the spirit of nominalism. Others argue that each resemblance relation is a particular, and is a resemblance relation simply in virtue of its resemblance to other resemblance relations. This generates an infinite regress, but many argue that it is not vicious
.
Another form of nominalism is trope theory. A trope is a particular instance of a property, like the specific greenness of a shirt. One might argue that there is a primitive, objective
resemblance relation that holds among like tropes. Another route is to argue that all apparent tropes are constructed out of more primitive tropes and that the most primitive tropes are the entities of complete physics
. Primitive trope resemblance may thus be accounted for in terms of causal indiscernibility. Two tropes are exactly resembling if substituting one for the other would make no difference to the events in which they are taking part. Varying degrees of resemblance at the macro level can be explained by varying degrees of resemblance at the micro level, and micro-level resemblance is explained in terms of something no less robustly physical than causal power. David Armstrong
, perhaps the most prominent contemporary realist, argues that such a trope-based variant of nominalism has promise, but holds that it is unable to account for the laws of nature in the way his theory of universals can.
Ian Hacking
has also argued that much of what is called social constructionism
of science in contemporary times is actually motivated by an unstated nominalist metaphysical view. For this reason, he claims, scientists and constructionists tend to "shout past each other".
and the philosophy of mathematics
should abstain from set theory
owes much to the writings of Nelson Goodman
(see especially Goodman 1977), who argued that concrete and abstract entities having no parts, called individuals exist. Collections of individuals likewise exist, but two collections having the same individuals are the same collection.
The principle of extensionality
in set theory assures us that any matching pair of curly braces enclosing one or more instances of the same individuals denote the same set. Hence {a,b}, {b,a}, {a,b,a,b} are all the same set. For Goodman and other nominalists, {a,b} is also identical to {a,{b}}, {b,{a,b}}, and any combination of matching curly braces and one or more instances of a and b, as long as a and b are names of individuals and not of collections of individuals. Goodman, Richard Milton Martin
, and Willard Quine all advocated reasoning about collectivities by means of a theory of virtual sets (see especially Quine 1969), one making possible all elementary operations on sets except that the universe
of a quantified variable cannot contain any virtual sets.
In the foundation of mathematics, nominalism has come to mean doing mathematics without assuming that sets in the mathematical sense exist. In practice, this means that quantified variables may range over universe
s of number
s, points
, primitive ordered pair
s, and other abstract ontological primitives, but not over sets whose members are such individuals. To date, only a small fraction of the corpus of modern mathematics can be rederived in a nominalistic fashion.
(entities) such as numbers and sets. Meanwhile, revolutionary nominalism is the project of replacing current scientific theories by alternatives dispensing with mathematical objects, see (Burgess, 1983, p. 96). A recent study extends the Burgessian critique to three nominalistic reconstructions: the reconstruction of analysis by Georg Cantor
, Richard Dedekind
, and Karl Weierstrass
that dispensed with infinitesimal
s; the constructivist
re-reconstruction of Weiertrassian analysis by Errett Bishop
that dispensed with the law of excluded middle
; and the hermeneutic reconstruction, by Carl Boyer, Judith Grabiner
, and others, of Cauchy's foundational contribution to analysis that dispensed with Cauchy's infinitesimals.
Metaphysics
Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...
view in philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
according to which general or abstract terms and predicates
Predicate (grammar)
There are two competing notions of the predicate in theories of grammar. Traditional grammar tends to view a predicate as one of two main parts of a sentence, the other being the subject, which the predicate modifies. The other understanding of predicates is inspired from work in predicate calculus...
exist, while universals
Universal (metaphysics)
In metaphysics, a universal is what particular things have in common, namely characteristics or qualities. In other words, universals are repeatable or recurrent entities that can be instantiated or exemplified by many particular things. For example, suppose there are two chairs in a room, each of...
or abstract object
Abstract object
An abstract object is an object which does not exist at any particular time or place, but rather exists as a type of thing . In philosophy, an important distinction is whether an object is considered abstract or concrete. Abstract objects are sometimes called abstracta An abstract object is an...
s, which are sometimes thought to correspond to these terms, do not exist. Thus, there are at least two main versions of nominalism. One version denies the existence of universals—things that can be instantiated or exemplified by many particular things (e.g. strength, humanity). The other version specifically denies the existence of abstract objects—objects that do not exist in space and time.
Most nominalists have held that only physical particulars in space and time are real, and that universals exist only post res, that is, subsequent to particular things. However, some versions of nominalism hold that some particulars are abstract entities (e.g. numbers), while others are concrete entities—entities that do exist in space and time (e.g. tables, chairs).
Nominalism is primarily a position on the problem of universals
Problem of universals
The problem of universals is an ancient problem in metaphysics about whether universals exist. Universals are general or abstract qualities, characteristics, properties, kinds or relations, such as being male/female, solid/liquid/gas or a certain colour, that can be predicated of individuals or...
, which dates back at least to Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
, and is opposed to realism—the view that universals do exist over and above particulars. However, the name "nominalism" emerged out of debates in medieval philosophy with Roscellinus
Roscellinus
Roscellinus, also called Roscelin of Compiègne or in Latin Roscellinus Compendiensis and Rucelinus , was a French philosopher and theologian, often regarded as the founder of nominalism .-Biography:...
. As John Stuart Mill
John Stuart Mill
John Stuart Mill was a British philosopher, economist and civil servant. An influential contributor to social theory, political theory, and political economy, his conception of liberty justified the freedom of the individual in opposition to unlimited state control. He was a proponent of...
once wrote, the early versions of nominalism were that "there is nothing general except names", hence the prefix "nomin-". This, however, is a more dated use of the term that is now considered to be a specific version of what is now called "nominalism".
History of nominalism
PlatoPlato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
was perhaps the first writer in Western philosophy to clearly state a non-Nominalist position, which he plainly endorsed:
...We customarily hypothesize a single form in connection with each of the many things to which we apply the same name. ... For example, there are many beds and tables. ... But there are only two forms of such furniture, one of the bed and one of the table. (Republic 596a-b, trans. Grube)
What about someone who believes in beautiful things, but doesn't believe in the beautiful itself…? Don't you think he is living in a dream rather than a wakened state? (Republic 476c)
The Platonic universals corresponding to the names "bed" and "beautiful" were the Form
Theory of Forms
Plato's theory of Forms or theory of Ideas asserts that non-material abstract forms , and not the material world of change known to us through sensation, possess the highest and most fundamental kind of reality. When used in this sense, the word form is often capitalized...
of the Bed and the Form of the Beautiful, or the Bed Itself and the Beautiful Itself. Platonic Forms were the first universals posited as such in philosophy.
Our term "universal" is due to the English translation of Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
's technical term katholou which he coined specially for the purpose of discussing the problem of universals. Katholou is a contraction of the phrase kata holou, meaning "on the whole".
Aristotle famously rejected certain aspects of Plato's Theory of Forms, but he clearly rejected Nominalism as well:
...'Man', and indeed every general predicate, signifies not an individual, but some quality, or quantity or relation, or something of that sort. (Sophistical Refutations xxii, 178b37, trans. Pickard-Cambridge)
The problem of universals
Nominalism arose in reaction to the problem of universalsProblem of universals
The problem of universals is an ancient problem in metaphysics about whether universals exist. Universals are general or abstract qualities, characteristics, properties, kinds or relations, such as being male/female, solid/liquid/gas or a certain colour, that can be predicated of individuals or...
, specifically accounting for the fact that some things are of the same type. For example, Fluffy and Kitzler are both cats, or, the fact that certain properties are repeatable, such as: the grass, the shirt
Shirt
A shirt is a cloth garment for the upper body. Originally an undergarment worn exclusively by men, it has become, in American English, a catch-all term for almost any garment other than outerwear such as sweaters, coats, jackets, or undergarments such as bras, vests or base layers...
, and Kermit the Frog
Kermit the Frog
Kermit the Frog is puppeteer Jim Henson's most famous Muppet creation, first introduced in 1955. He is the protagonist of many Muppet projects, most notably as the host of The Muppet Show, and has appeared in various sketches on Sesame Street, in commercials and in public service announcements over...
are green. One wants to know in virtue of what are Fluffy and Kitzler both cats, and what makes the grass, the shirt, and Kermit green.
The realist
Philosophical realism
Contemporary philosophical realism is the belief that our reality, or some aspect of it, is ontologically independent of our conceptual schemes, linguistic practices, beliefs, etc....
answer is that all the green things are green in virtue of the existence
Existence
In common usage, existence is the world we are aware of through our senses, and that persists independently without them. In academic philosophy the word has a more specialized meaning, being contrasted with essence, which specifies different forms of existence as well as different identity...
of a universal; a single abstract
Abstraction
Abstraction is a process by which higher concepts are derived from the usage and classification of literal concepts, first principles, or other methods....
thing, in this case, that is a part of all the green things. With respect to the color of the grass, the shirt and Kermit, one of their parts is identical. In this respect, the three parts are literally one. Greenness is repeatable because there is one thing that manifests
Exemplification
Exemplification is a mode of symbolization characterized by the relation between a sample and what it refers to.- Description :Unlike ostension, which is the act of showing or pointing to a sample, exemplification is possession of a property plus reference to its label...
itself wherever there are green things.
Nominalism denies the existence of universals. The motivation for this flows from several concerns, the first one being where they might exist. Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
famously held, on one interpretation, that there is a realm of abstract forms or universals apart from the physical world (see theory of the forms). Particular physical objects merely exemplify or instantiate the universal. But this raises the question: Where is this universal realm? One possibility is that it is outside of space and time. A view sympathetic with this possibility holds that, precisely because some form is immanent in several physical objects, it must also transcend each of those physical objects; in this way, the forms are "transcendant" only insofar as they are "immanent" in many physical objects. In other words, immanence implies transcendence; they are not opposed to one another. (Nor, on this view, would there be a separate "world" or "realm" of forms that is distinct from the physical world, thus shirking much of the worry about where to locate a "universal realm".) However, naturalists
Metaphysical naturalism
Metaphysical naturalism, also called ontological naturalism and philosophical naturalism, or just naturalism, is a philosophical worldview and belief system that holds that there is nothing but natural elements, principles, and relations of the kind studied by the natural sciences, i.e., those...
assert that nothing is outside of space and time. Some Neoplatonists, such as the pagan philosopher Plotinus
Plotinus
Plotinus was a major philosopher of the ancient world. In his system of theory there are the three principles: the One, the Intellect, and the Soul. His teacher was Ammonius Saccas and he is of the Platonic tradition...
and the Christian philosopher St. Augustine
Augustine of Hippo
Augustine of Hippo , also known as Augustine, St. Augustine, St. Austin, St. Augoustinos, Blessed Augustine, or St. Augustine the Blessed, was Bishop of Hippo Regius . He was a Latin-speaking philosopher and theologian who lived in the Roman Africa Province...
, imply (anticipating conceptualism
Conceptualism
Conceptualism is a philosophical theory that explains universality of particulars as conceptualized frameworks situated within the thinking mind. Intermediate between Nominalism and Realism, the conceptualist view approaches the metaphysical concept of universals from a perspective that denies...
) that universals are contained within the mind of God
Preformation theory
The Preformation theory states that "knowledge is possible only because God has endowed humans with certain innate ideas along with dispositions or aptitudes in certain ways." This was recognized by Immanuel Kant as an alternative to his theory regarding the categories of understanding and their...
. To complicate things, what is the nature of the instantiation
Instantiation principle
The "Principle of Instantiation" or "Principle of Exemplification" is a thesis in philosophy that states that there can be no uninstantiated or unexemplified properties . In other words, it is impossible for a property to exist which is not had by some object. Aristotle is well-known for...
or exemplification
Exemplification
Exemplification is a mode of symbolization characterized by the relation between a sample and what it refers to.- Description :Unlike ostension, which is the act of showing or pointing to a sample, exemplification is possession of a property plus reference to its label...
relation?
Conceptualists
Conceptualism
Conceptualism is a philosophical theory that explains universality of particulars as conceptualized frameworks situated within the thinking mind. Intermediate between Nominalism and Realism, the conceptualist view approaches the metaphysical concept of universals from a perspective that denies...
hold a position intermediate between nominalism and realism
Philosophical realism
Contemporary philosophical realism is the belief that our reality, or some aspect of it, is ontologically independent of our conceptual schemes, linguistic practices, beliefs, etc....
, saying that universals exist only within the mind
Mind
The concept of mind is understood in many different ways by many different traditions, ranging from panpsychism and animism to traditional and organized religious views, as well as secular and materialist philosophies. Most agree that minds are constituted by conscious experience and intelligent...
and have no external or substantial reality.
Moderate realists
Moderate realism
Moderate realism is a position in the debate on the metaphysics of universals which holds that there is no realm in which universals exist , nor do they really exist within the individuals as universals, but rather universals really exist within the particulars as individualised, and multiplied...
hold that there is no realm in which universals exist, but rather universals are located in space and time wherever they are manifest. Now, recall that a universal, like greenness, is supposed to be a single thing. Nominalists consider it unusual that there could be a single thing that exists in multiple places simultaneously. The realist maintains that all the instances of greenness are held together by the exemplification relation, but this relation cannot be explained.
Finally, many philosophers prefer simpler ontologies
Ontology
Ontology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
populated with only the bare minimum of types of entities, or as W. V. Quine said "They have a taste for 'desert landscapes.'" They attempt to express everything that they want to explain without using universals such as "catness" or "chairness."
Varieties of nominalism
There are various forms of nominalism ranging from extreme to almost-realist. One extreme is predicate nominalism, which states that Fluffy and Kitzler, for example, are both cats simply because the predicate 'is a cat' applies to both of them. And this is the case for all similarity of attribute among objects. The main criticism of this view is that it does not provide a sufficient solution to the problem of universals. It fails to provide an account of what makes it the case that a group of things warrant having the same predicate applied to themResemblance nominalists believe that 'cat' applies to both cats because Fluffy and Kitzler resemble an exemplar
Exemplar
Exemplar, in the sense developed by philosopher of science Thomas Kuhn, is a well known usage of a scientific theory.According to Kuhn, scientific practice alternates between periods of normal science and extraordinary/revolutionary science...
cat closely enough to be classed together with it as members of its kind
Natural kind
In philosophy, a natural kind is a "natural" grouping, not an artificial one. Or, it is something that a set of things has in common which distinguishes it from other things as a real set rather than as a group of things arbitrarily lumped together by a person or group of people.If any natural...
, or that they differ from each other (and other cats) quite less than they differ from other things, and this warrants classing them together. Some resemblance nominalists will concede that the resemblance relation is itself a universal, but is the only universal necessary. This betrays the spirit of nominalism. Others argue that each resemblance relation is a particular, and is a resemblance relation simply in virtue of its resemblance to other resemblance relations. This generates an infinite regress, but many argue that it is not vicious
Virtuous circle and vicious circle
A virtuous circle and a vicious circle are economic terms. They refer to a complex of events that reinforces itself through a feedback loop. A virtuous circle has favorable results, while a vicious circle has detrimental results...
.
Another form of nominalism is trope theory. A trope is a particular instance of a property, like the specific greenness of a shirt. One might argue that there is a primitive, objective
Objectivity (science)
Objectivity in science is a value that informs how science is practiced and how scientific truths are created. It is the idea that scientists, in attempting to uncover truths about the natural world, must aspire to eliminate personal biases, a priori commitments, emotional involvement, etc...
resemblance relation that holds among like tropes. Another route is to argue that all apparent tropes are constructed out of more primitive tropes and that the most primitive tropes are the entities of complete physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
. Primitive trope resemblance may thus be accounted for in terms of causal indiscernibility. Two tropes are exactly resembling if substituting one for the other would make no difference to the events in which they are taking part. Varying degrees of resemblance at the macro level can be explained by varying degrees of resemblance at the micro level, and micro-level resemblance is explained in terms of something no less robustly physical than causal power. David Armstrong
David Malet Armstrong
David Malet Armstrong , often D. M. Armstrong, is an Australian philosopher. He is well-known for his work on metaphysics and the philosophy of mind, and for his defence of a factualist ontology, a functionalist theory of the mind, an externalist epistemology, and a necessitarian conception of the...
, perhaps the most prominent contemporary realist, argues that such a trope-based variant of nominalism has promise, but holds that it is unable to account for the laws of nature in the way his theory of universals can.
Ian Hacking
Ian Hacking
Ian Hacking, CC, FRSC, FBA is a Canadian philosopher, specializing in the philosophy of science.- Life and works :...
has also argued that much of what is called social constructionism
Social constructionism
Social constructionism and social constructivism are sociological theories of knowledge that consider how social phenomena or objects of consciousness develop in social contexts. A social construction is a concept or practice that is the construct of a particular group...
of science in contemporary times is actually motivated by an unstated nominalist metaphysical view. For this reason, he claims, scientists and constructionists tend to "shout past each other".
Analytic philosophy and mathematics
The notion that philosophy, especially ontologyOntology
Ontology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
and the philosophy of mathematics
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
should abstain from set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
owes much to the writings of Nelson Goodman
Nelson Goodman
Henry Nelson Goodman was an American philosopher, known for his work on counterfactuals, mereology, the problem of induction, irrealism and aesthetics.-Career:...
(see especially Goodman 1977), who argued that concrete and abstract entities having no parts, called individuals exist. Collections of individuals likewise exist, but two collections having the same individuals are the same collection.
The principle of extensionality
Extensionality
In logic, extensionality, or extensional equality refers to principles that judge objects to be equal if they have the same external properties...
in set theory assures us that any matching pair of curly braces enclosing one or more instances of the same individuals denote the same set. Hence {a,b}, {b,a}, {a,b,a,b} are all the same set. For Goodman and other nominalists, {a,b} is also identical to {a,{b}}, {b,{a,b}}, and any combination of matching curly braces and one or more instances of a and b, as long as a and b are names of individuals and not of collections of individuals. Goodman, Richard Milton Martin
Richard Milton Martin
Richard Milton Martin was an American logician and analytic philosopher. In his Ph.D. thesis written under Frederic Fitch, Martin discovered virtual sets a bit before Quine, and was possibly the first non-Pole other than Joseph Henry Woodger to employ a mereological system...
, and Willard Quine all advocated reasoning about collectivities by means of a theory of virtual sets (see especially Quine 1969), one making possible all elementary operations on sets except that the universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...
of a quantified variable cannot contain any virtual sets.
In the foundation of mathematics, nominalism has come to mean doing mathematics without assuming that sets in the mathematical sense exist. In practice, this means that quantified variables may range over universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...
s of number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s, points
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
, primitive ordered pair
Ordered pair
In mathematics, an ordered pair is a pair of mathematical objects. In the ordered pair , the object a is called the first entry, and the object b the second entry of the pair...
s, and other abstract ontological primitives, but not over sets whose members are such individuals. To date, only a small fraction of the corpus of modern mathematics can be rederived in a nominalistic fashion.
Critique of mathematical nominalism
A critique of nominalist reconstructions in mathematics was undertaken by Burgess (1983) and Burgess and Rosen (1997). Burgess distinguished two types of nominalist reconstructions. Thus, hermeneutic nominalism is the hypothesis that science, properly interpreted, already dispenses with mathematical objects(entities) such as numbers and sets. Meanwhile, revolutionary nominalism is the project of replacing current scientific theories by alternatives dispensing with mathematical objects, see (Burgess, 1983, p. 96). A recent study extends the Burgessian critique to three nominalistic reconstructions: the reconstruction of analysis by Georg Cantor
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...
, Richard Dedekind
Richard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
, and Karl Weierstrass
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
that dispensed with infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
s; the constructivist
Constructivism (mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its...
re-reconstruction of Weiertrassian analysis by Errett Bishop
Errett Bishop
Errett Albert Bishop was an American mathematician known for his work on analysis. He is the father of constructive analysis, because of his 1967 Foundations of Constructive Analysis, where he proved most of the important theorems in real analysis by constructive methods.-Life:Errett Bishop's...
that dispensed with the law of excluded middle
Law of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
; and the hermeneutic reconstruction, by Carl Boyer, Judith Grabiner
Judith Grabiner
Judith Victor Grabiner is an American mathematician who is among the best-known historians of mathematics. She earned her B.S. in mathematics at the University of Chicago in 1960 and her M.A. and Ph.D. in the history of science at Harvard University, and is currently Flora Sanborn Pitzer...
, and others, of Cauchy's foundational contribution to analysis that dispensed with Cauchy's infinitesimals.
See also
- AbstractionAbstractionAbstraction is a process by which higher concepts are derived from the usage and classification of literal concepts, first principles, or other methods....
- Abstract objectAbstract objectAn abstract object is an object which does not exist at any particular time or place, but rather exists as a type of thing . In philosophy, an important distinction is whether an object is considered abstract or concrete. Abstract objects are sometimes called abstracta An abstract object is an...
- ConceptualismConceptualismConceptualism is a philosophical theory that explains universality of particulars as conceptualized frameworks situated within the thinking mind. Intermediate between Nominalism and Realism, the conceptualist view approaches the metaphysical concept of universals from a perspective that denies...
- Concrete (philosophy)
- IdeaIdeaIn the most narrow sense, an idea is just whatever is before the mind when one thinks. Very often, ideas are construed as representational images; i.e. images of some object. In other contexts, ideas are taken to be concepts, although abstract concepts do not necessarily appear as images...
- ObjectObject (philosophy)An object in philosophy is a technical term often used in contrast to the term subject. Consciousness is a state of cognition that includes the subject, which can never be doubted as only it can be the one who doubts, and some object or objects that may or may not have real existence without...
- Problem of universalsProblem of universalsThe problem of universals is an ancient problem in metaphysics about whether universals exist. Universals are general or abstract qualities, characteristics, properties, kinds or relations, such as being male/female, solid/liquid/gas or a certain colour, that can be predicated of individuals or...
- Psychological nominalismPsychological nominalismPsychological nominalism is the view advanced in Wilfrid Sellars' paper "Empiricism and Philosophy of Mind" that explains psychological concepts in terms of public language use...
- Realism (philosophy)
- Substantial formSubstantial formA theory of substantial forms asserts that forms organize matter and make it intelligible. Substantial forms are the source of properties, order, unity, identity, and information about objects....
- Universal (metaphysics)Universal (metaphysics)In metaphysics, a universal is what particular things have in common, namely characteristics or qualities. In other words, universals are repeatable or recurrent entities that can be instantiated or exemplified by many particular things. For example, suppose there are two chairs in a room, each of...
- William of OckhamWilliam of OckhamWilliam of Ockham was an English Franciscan friar and scholastic philosopher, who is believed to have been born in Ockham, a small village in Surrey. He is considered to be one of the major figures of medieval thought and was at the centre of the major intellectual and political controversies of...
External links
- Nominalism, Realism, Conceptualism, from The Catholic Encyclopedia.
- Nominalism, from the BELIEVE Religious Information Collection.
- Rose, Burgess: Nominalism Reconsidered in The Oxford handbook of philosophy of mathematics and logic
- Nominalism in Metaphysics from The Stanford Encyclopedia of Philosophy
- Richard Utz and Terry Barakat, "Medieval Nominalism and the Literary Questions: Selected Studies." Perspicuitas