NUMBER

Numbers are central to science. They underlie what Galileo Galilei and Isaac Newton called the primary properties of things, the properties that can be measured (John Locke listed these as number, motion and rest, size, figure, and impenetrability). These underlie secondary properties (like colors and musical harmonies and discords), which in turn underlie the tertiary properties, like beauty, which make life worth living.

The centrality of numbers to science indirectly confers on them philosophical significance, but they have also played a direct role in metaphysics. Plato's theory of universals begins from the problem of the One over Many. Behind the superficial diversity of things in the world, it is often the case that there is one thing that many numerically distinct individuals share in common. For instance, when one doubles the length of the string on a lyre or the length of a column of air in a flute, the note it sounds is always lowered by the same musical interval, an octave. The things that distinct individuals share in common are called universals, and "Platonism" is used as a name for a broad and loose family of theories that affirm the existence of universals.

The existence of numbers has always been central to the history of Platonism, from ancient times to the present. In the nineteenth and twentieth centuries foundational work in the philosophy of mathematics, especially by Gottlob Frege and Bertrand Russell, affirmed the existence of numbers. Following them, Willard Van Orman Quine affirmed the existence of numbers, and he rightly called this doctrine "Platonism". Quine argued that it is reasonable to believe in the existence of numbers because numbers are central to mathematics, which in turn is central to science. This reason for believing in numbers is close to the guiding Pythagorean and Platonist idea that to understand the world we must find the unified mathematical patterns that lie behind the diversities of appearances.

Early History of Numbers

The history of numbers in India, China, and elsewhere is deep and diverse, but it is still not properly understood. In ancient and modern histories of ideas in Europe, the origin of geometry was traditionally traced to ancient Egypt; and relatively sophisticated advances in arithmetic and algebra have been recognized as having emerged in Mesopotamia; and both these sources entered European traditions through ancient Greece. Knowledge of this ancient history is improving, but it is still incomplete.

The early mathematical advances of ancient Greece are better known, though even here the evidence is sparse. Almost no written records survive from the Pythagorean oral traditions before Plato. What survives from before Euclid's Elements consists in little more than hints in Plato and Aristotle.

Euclid's Elements, the first systematic presentation of geometry and arithmetic, is magnificent, but little is known of its sources and motivations. It is relatively apparent, however, that some of his theorems consist in translations of algebraic results, known in Mesopotamia, into geometric counterparts. For instance, an algebraic thesis, like (a + b )² = (a ² + 2ab + b ²), would become a theorem concerned with the division of a square into two smaller squares and two rectangles. For some reason, the mathematicians of Plato's Academy emphasized geometry rather than arithmetic, and arithmetic was subsumed under geometry.

Proliferation of Kinds of Numbers

Besides the whole numbers (or natural numbers) the Greeks also recognized relationships of ratio between numbers. For example, the numbers 9 and 6 stand in the same ratio as 3 to 2, and one can call this ratio (3:2). This same relationship of ratio that holds between any two numbers will also hold between two possible geometrical lengths.

However, among the relationships of ratio that hold between various magnitudes, as for instance between lengths of lines, there are some that do not hold between any two whole numbers. Plato and Aristotle allude, many times, to a proof that no ratio between whole numbers will match the relationship of proportion that holds between the diagonal and the side of a perfect square. This fact would now be expressed by saying that √2 is an irrational number, which means that there are no whole numbers a and b such that a / b = √2.

The ancient Greeks thought of ratios among lines as forming a domain distinct from the domain of numbers. Numbers consisted simply of whole numbers. As the centuries advanced, the term number gradually expanded to include the entire domain of what are now called the positive real numbers. This domain includes all the irrational numbers, such as √2 and π, that can be represented as nonterminating decimals (such as 3.1415926 …). This domain includes, as a subdomain, the rational numbers, which are the ratios that hold between whole numbers; and, as a smaller subdomain, the integers, which correspond to just the rational ratios to the unit measure. The domain of number did not initially, however, include the number zero or the negative numbers.

Over several centuries the domain of things that were included as numbers expanded to include zero and negative numbers. First, there was an expansion to include a symbol "0" that was at first to be thought of not as signifying any number, but just as a place holder in the system of Arabic notation that is used today. In the notation "12," the "1" is placed in the second column from the right, and this means that it signifies one group of ten. Take 2 away from 12 and the result is written "10" with the "0" not referring to anything at all, but just serving to keep the "1" in the second column, so that it continues to signify one group of ten.

As time went by, however, the symbol "0" did come to be thought of as standing for something that might be called "the number zero", trusting that there was some suitable thing for this symbol to refer to. It was only gradually that any clear conception began to arise of what kind of thing this number zero might be.

Likewise, negative numbers began as notation that did not refer to any extra numbers, but just told one what to do with ordinary, positive whole numbers. With time, however, this notation came to be thought of as referring to new numbers, and eventually a conception emerged about what kinds of things these new numbers might be.

There was also a tentative expansion, with deep philosophical misgivings, to include what are now known as imaginary and complex numbers. Briefly, the imaginary number i —assuming there is such a thing, and calling it a number—is defined to be that mathematical object that is such that the ratio of 1 to it is the same as the ratio of it to minus 1. That is, i / 1 = −1 / i, so that i ² = −1. Complex numbers consist of all the numbers that can be obtained from i by taking multiples of it and adding the result to other numbers.

There was also a tentative expansion, with deep philosophical misgivings, to include infinitesimal magnitudes. These extra entities seemed to be indispensable in the new mathematical theory of physical magnitudes like velocity and acceleration, invented by Newton and Gottfried Wilhelm Leibniz and referred to as "calculus" or, in its most general form, "analysis".

Despite the immense success of the calculus in science, the concept of an infinitesimal—a magnitude greater than zero, but less than any finite magnitude—was viewed with some suspicion. In the nineteenth century, through the work of Augustine-Louis Cauchy and Karl Weierstrass, the concept of an infinitesimal was replaced by the concept of the limit of a sequence of numbers. An infinite sequence of numbers s₁, s₂, s₃, … approaches the limit l if the difference between l and s_n can be made as small as one likes by taking sufficiently large values of n. That is, given any positive number d, no matter how small, there is some number N such that the difference between l and s_n is less than d, for every n (N.

Using this concept, the nineteenth-century mathematicians showed how the concepts of continuity, convergence, differential, and integral could all be precisely defined. In this way, it was shown how talk of infinitesimals could be dispensed with entirely.

A further nineteenth-century development was the introduction by Georg Cantor of the concept of a transfinite number. The transfinite numbers can be thought of as measuring the size of infinite sets. Cantor introduced the symbol ℵ₀ (pronounced "aleph-null") for the number measuring the size of the set of all positive whole numbers and the symbol c for the transfinite number measuring the size of the set of all real numbers. By a simple, yet ingenious argument (the celebrated diagonal argument), Cantor was able to show that there are more real numbers than whole numbers: c > ℵ₀.

Cantor proved that there are always more subsets of a given set than elements of that set (so there are more sets of natural numbers than natural numbers for example). Hence, given any transfinite number measuring the size of an infinite set, there is a larger transfinite number, which measures the size of the set of all subsets of that set.

Cantor developed a transfinite arithmetic for these new numbers, showing how operations corresponding to addition and exponentiation could be defined for them. Again, the new numbers were viewed initially with the deepest suspicion by the mathematical community.

Frege and the Paradoxes

The work of the nineteenth-century mathematicians had begun a reverse process of defining one kind of number in terms of simpler kinds. The complex numbers, it had been shown, could be defined as pairs of real numbers (like the x-y coordinates of Cartesian geometry) along with special rules for adding and multiplying these pairs. The real numbers had been shown by Julius Dedekind and Cauchy to be definable as infinite sequences or sets of rational numbers, while the rational numbers themselves can be identified with sets of pairs of natural numbers.

What of the natural numbers themselves? Frege's work can be seen as an attempt to complete this reverse process of rigorization by providing a firm foundation for the fundamental theory of the natural numbers. Dedekind and Giuseppe Peano had independently specified some simple axioms for that theory (called number theory or arithmetic). However, Frege wanted to answer the questions: What are the natural numbers? How may they be defined? The Dedekind-Peano axioms specify the laws governing the numbers, but do not provide a definition of them.

Imagine if one thought that the number of soldiers in an army was one of the army's most significant properties. One might then think of whole numbers as properties of aggregates. However, as Frege pointed out, if one points to the things on a desk and asks how many there are, one has not yet asked a complete question. There may be two decks of cards; and if so, then there are also 106 cards; and there are a great many molecules; and so on.

This suggests that number is a property of properties. The property of "being a deck of cards on the table" has the property of having one instance; the property of "being a card on the table" has the property of having 106 instances; and so on. The property of "being a unicorn" has the property of having no instances. That higher-order property, the property of having no instances, might aptly be called the number zero.

Consider, then, the theory that the number 2 is a property of a property, namely the property of having two instances, that the number 3 is the property of having three instances, and so on. Frege turned decisively aside from this theory. He argued that numbers could not be universals or concepts, but had to be objects.

For Frege, the fundamental kind of expression used to ascribe numbers to things are expressions like "the number of cards on the desk" or "the number of planets in the solar system." The expression "the number of Fs" is a singular term, purporting to pick out an object, in just the same way as "the brother of John" is a singular term, purporting to pick out a certain individual. So for Frege, ascriptions of number depend for their truth on the existence of objects, which are the referents of expressions of the form "the number of Fs."

Is it legitimate to suppose that given any general term F, there is also an object corresponding to the "the number of Fs"? Frege held that it is legitimate to speak of objects of a certain kind, provided there is a criterion of identity for them. What is the criterion of identity for numbers? The answer is given by the following principle, known as Hume's principle: "The number of Fs = the number of Gs if and only if (iff) there is a one-to-one correspondence between the Fs and the Gs."

A one-to-one correspondence is a relation that pairs each F with exactly one G and each G with exactly one F. So, for example, the number of knives on the table is equal to the number of forks provided that each fork can be paired with a unique knife and each knife with a unique fork.

Frege demonstrated that all the Dedekind-Peano axioms for number theory can be proved from Hume's principle alone, given appropriate definitions that he devised; a fact now known as Frege's theorem.

Frege attempted to go further by giving an explicit definition of "the number of Fs," from which Hume's principle itself could be proved. He defined "the number of Fs" as the set of all properties that can be put in one-to-one correspondence with the Fs. That is, the number n is identified with the extension of the second-order property of having n -members.

This was a disaster. The principle concerning sets that Frege appealed to in his derivation of Hume's principle states that every predicate has an extension. The extension of a predicate is the set of all (and only) those objects that satisfy the predicate. As Russell's paradox shows, however, this principle is inconsistent. If every predicate has an extension, then the predicate "is not a member of itself" has an extension, which would be the set of all (and only) the objects that are not members of themselves. Call this set R. It follows that R is a member of R iff R is not a member of R, a contradiction. Frege's logical system had turned out to be inconsistent. This was the first of a number of paradoxes of set theory that were to have a formative influence on subsequent work in the foundations of mathematics.

There were varying responses to Russell's paradox. Russell and Alfred North Whitehead took one approach: the theory of types. Ernst Zermelo and others took a different approach: that of axiomatic set theory. Given the now standard axioms for set theory, the Frege-Russell definition of the numbers will not work; the assumption that there is a nonempty set of all three-membered sets, for example, leads to a contradiction. A different approach to the definition of the numbers is required. Instead of taking the numeral n to refer to the set of all n -membered sets, it can be taken to refer to some particular, paradigm example of an n -membered set.

John von Neumann provided an effective sequence of paradigm n -membered sets. The number zero is the paradigm zero-membered set: the empty set, ∅. The number 1 is the set whose only member is zero. The number 2 is the paradigm two-membered set whose members are 0 and 1. And in general, each number n is the n -membered set whose members consist of all and only the whole numbers from 0 up to (n − 1).

One can then say that there are n members of a particular set iff that set can be placed into a one-to-one correlation with the paradigm n -membered set. For instance, there are two decks of cards on the table iff the members of the set of decks of cards on the table can be placed into a one-to-one correspondence with the members of the paradigm two-membered set {∅, {∅}}.

Philosophies of Number

Philosophical accounts of number (and mathematics more generally) can be divided into two broad categories: realist and antirealist.

A realist about number holds that statements concerning numbers are objectively true or false. On this view, statements such as "there are nine planets in the solar system," "there are infinitely many prime numbers," "34957 + 70764 = 105621," or "every even number greater than two is the sum of two primes" (Goldbach's conjecture), say something that is objectively either true or false, even if no one knows which it is. In addition, the realist claims that some such statements are indeed true. That is, the realist typically accepts as true most, or all, of accepted mathematics.

By contrast, an antirealist denies one or both of the two realist claims. That is, the antirealist will deny that there is an objective fact of the matter about the truth value of all statements concerning number or that all currently accepted mathematical statements concerning number are actually true.

The Argument for Platonism

Platonism, as that term is used in modern philosophy of mathematics, is the view that mathematics is the study of an objective realm of independently existing objects. In addition, the platonist holds that these objects are abstract, rather than physical objects. A physical object is something that (if it exists) has a location in space and time, can undergo changes of state, and can interact causally with other spatiotemporally located objects. Cups and saucers, stars and planets, plants and animals, and atoms and photons are all examples of physical objects. By contrast, an abstract object is something that (if it exists) lacks some or all of these properties. Abstract objects have no location in space and time, they have no state and no history, and they do not interact causally with other objects.

The main philosophical argument for platonism in modern philosophy proceeds as follows. Many statements of arithmetic appear to make existential claims. For example, the statement "there is a prime number greater than three" asserts the existence of an object having certain properties. Since many such arithmetical statements are true, it follows that numbers exist. This does not yet show that numbers must be abstract, but various arguments can be given against the alternatives. For example, every physical object has a location in space and exists for a certain time. Numbers have neither of these properties. Then again, there are infinitely many numbers, but perhaps a finite number of physical objects. It follows that numbers, if they exist, must be nonphysical, abstract objects.

The argument for platonism can be summarized as follows:

P1. Arithmetical sentences express statements that are objectively true or false

P2. Some arithmetical statements are true

P3. Arithmetical statements quantify over certain objects (numbers)

Therefore:

C1: Numbers exist.

However:

P4: Numbers, if they exist, must be abstract (nonphysical, nonmental) objects.

Therefore:

C: Numbers are abstract objects.

The Epistemological Problem

The central problem facing a platonist philosophy of number is epistemological. Abstract objects cannot be directly perceived, nor can they have any effects on objects or processes that can be directly perceived. How then is it possible for us to know anything at all about such objects? The causal isolation of abstract objects appears to make them unknowable. Hence either we have no mathematical knowledge, or platonism is not the correct view of mathematics.

Before Frege, philosophers had postulated that human beings have some kind of direct cognitive access to mathematical objects, through perception or some rational faculty analogous to perception, or (for Immanuel Kant) through an a priori intuition or construction.

According to Frege our our only access to numbers is through our knowledge of the truth-values of arithmetical statements. Certain sentences of our language contain terms standing for numbers and quantifiers that range over numbers. If it can be shown that some of those sentences are true (and Frege hoped to show that they are logically true), then we will have explained how we can know about numbers, even though we have no direct perceptual or causal contact with them.

If the reduction of arithmetic to logic could be carried out, our knowledge of numbers would have been shown to be based on our knowledge of the truth of the basic laws of logic. Frege thought there was no real problem about how we know that the laws of logic are true; we can just see that they are. Explaining the psychological mechanisms that give us this ability is outside the scope of philosophy and can be left to the psychologists. The discovery of the paradoxes ruined this comfortable picture, showing that we have no infallible insight into the fundamental truths of logic after all. The reduction of arithmetic to set theory does not resolve this problem. Our knowledge of the basic laws of set theory cannot be any more certain or secure than our knowledge of the fundamental laws of arithmetic.

Realist Alternatives to Platonism

In view of the epistemological problem for platonism, many philosophers of mathematics have sought to avoid the conclusion that numbers are abstract objects. However, if that conclusion should be rejected, the argument for platonism given earlier must be unsound. Alternatives to platonism can be usefully classified according to which of the premises of that argument are rejected.

An obvious point at which that argument might be attacked is at premise P4. That is, one could deny that numbers, if they exist, must be abstract. Along these lines are various attempts to provide a physicalist account of mathematical objects such as numbers and sets. Such accounts differ from platonism only in denying that numbers are entirely nonphysical and abstract. The payoff is epistemological. If, for example, numbers are properties or relations that can be instantiated by ordinary physical objects, then some basic knowledge of numbers could be acquired by ordinary perception.

Another realist alternative is to accept P1 and P2, but deny P3; the claim that mathematical statements quantify over a domain of special objects of some kind. One strategy is to think of arithmetic as the theory, not of a special realm of objects, but of a certain pattern or structure. In the case of arithmetic the structure in question is that shared by any infinite progression of objects (also called an ω-sequence) in which (1) there is a unique first element and (2) for any given element there is a distinct, unique next element in the sequence, called the successor of the given element.

According to one variety of structuralism, the truths of arithmetic are simply those that hold in every system of objects that form an ω-sequence. An equation such as 2 + 1 = 3 is interpreted as elliptical for the generalization; "If S is any system of objects that form an ω-sequence, then the successor of the successor of the first element of S added to the successor of the first element of S is equal to the successor of the successor of the successor of the first element of S."

Structuralism is often motivated by a certain ontological problem for platonism. According to the platonist, sets and numbers are abstract objects. What is the relationship between them? We have already described one way in which the natural numbers can be defined as sets. This is the definition of the natural numbers as the von Neumann numbers: 0 = ∅, 1 = {0}, 2 = {0,1}, and 3 = {0,1,2} and in general, N +1 = {0, 1, …, N }.This is not the only possible set-theoretic definition of the natural numbers, however. Zermelo, for example, defined the sequence as follows: 0 = ∅, 1 = {0}, 2 = {1}, and 3 = {2} and in general, N+1 = {N}.

From a purely mathematical point of view the definitions seem equally valid, since they both validate exactly the same theorems of arithmetic. However, the two definitions are certainly not equivalent, since they identify some numbers with distinct sets; on von Neumann's definition 2 = {∅, {∅}} (a set with two members), while on Zermelo's definition 2 = {{∅}} (a set with just one member).

From a platonist perspective there is something puzzling about this. If numbers are independently existing objects, then there must be a fact of the matter about which set, if any, the number 2 is identical with. It cannot be that there are two equally correct definitions of the number 2 that identify it with different sets, but this is exactly what one seems to have in the case of the von Neumann and Zermelo definitions: two equally correct accounts of the number 2 that assign it to distinct sets.

A generalization of this line of argument yields the conclusion that numbers cannot be objects of any kind. Any definition of numbers in terms of particular mathematical objects of some other kind is arbitrary, in the sense that equivalent, but distinct alternative definitions will always be available. However, if, as the platonist holds, numbers are objects, there must be a fact of the matter about which objects the numbers really are. So there must be something wrong with platonism.

From a structuralist perspective, however, this kind of ontological relativity is readily explicable. For on that account, arithmetic is not concerned with a domain of specific objects, but only with what holds good in all ω-sequences. The sequences of sets defined by von Neumann and Zermelo are both examples of ω-sequences, so both systems have the required structure. Any system of objects (sets or otherwise) having the same structure will do just as well, for arithmetic is just the theory of the properties shared by all ω-sequences.

Conventionalism

A different approach, long popular with empiricists, is to say that mathematics is concerned not with objects, but with relations between concepts. A good example is the account of mathematics associated with the philosophical movement known as logical positivism, which had its heyday in the 1930s and 1940s. According to the positivists, the truths of logic and mathematics are alike in being analytic, by which they meant that they are true solely in virtue of the meanings of the symbols they contain, meanings that are established by linguistic convention.

On this view, 2 + 2 = 4 is true because of the stipulations we have laid down governing the use of the symbols "2," "4," "+," and "=." As such it is completely without empirical content and this explains the irrelevance of empirical evidence to mathematics. No fact about the world can contradict the statement 2 + 2 = 4, because its truth does not depend on facts about the world, but only on facts about what the mathematical symbols occurring in it mean. What our symbols mean is a matter of arbitrary linguistic convention. We can simply stipulate that our symbols are going to have certain meanings and then the truth of various statements involving them will follow. The truths of arithmetic on this view are records of the stipulations we have laid down governing the use of the arithmetical symbols.

Largely as a result of criticisms developed by Quine and others, conventionalism is no longer widely accepted. One difficulty is that even if it were possible simply to stipulate that the terms of a mathematical theory are to be assigned whatever meaning makes all the axioms turn out true, the stipulation will backfire if the axioms are inconsistent. Whether the axioms are consistent or not is itself a mathematical fact which is independent of our stipulations and conventions. If so, then not all mathematical facts can be purely conventional or true in virtue of meaning.

The specific objections to conventionalism are however, less significant than the alternative account of the epistemology of mathematics developed by Quine, which if correct, would undermine the main epistemological motivation for conventionalism. Quine's alternative account is described in the final section of this article.

Nominalism

Nominalism is the philosophical thesis that there are no abstract objects that is, everything that exists is a concrete, physical particular. In interpreting mathematics and science then, the nominalist has two options. One option is to say that despite appearances, mathematical and scientific theories do not involve reference to abstract objects after all. The other option is to say that they do and are therefore literally false. The nominalist may then seek to provide a positive account of mathematical and scientific theories, showing how they can be reformulated so as to avoid any reference to abstract objects. The result would be an error theory of science and mathematics.

This second approach is the one taken by the nominalist philosopher Hartry Field, who has attempted to demonstrate that reference to abstract objects can be eliminated from science by showing how nominalistic versions of physical theories might be constructed: versions which do not presuppose the existence of abstract objects such as numbers or functions. The interested reader is referred to the bibliography for further details of the construction and the philosophical debate surrounding it.

Formalism

This type of nominalist antirealism concerning numbers and other abstract objects consists in a denial only of the second premise (P2) of the argument for platonism given earlier. On such a view, mathematical sentences express statements which can be true or false, but it is argued that there is no good reason for thinking that any mathematical statements involving quantification over abstract objects are literally true.

More radical types of antirealism deny the first premise (P1) of the argument for platonism. One way of denying that premise is to say that mathematical sentences do not express statements that could be true or false at all, objectively or otherwise. This is the approach taken by the formalist account of mathematics. On this view, mathematics is not a body of statements that can be true or false. Instead, mathematics is thought of as analogous to a game, like chess. It is a game played with symbols according to certain rules.

The most sophisticated version of the formalist account of mathematics is that proposed by the mathematician David Hilbert in the 1920s and early 1930s. According to Hilbert mathematics has a meaningful part and a purely formal part. The meaningful part consists of finitary statements. These are decidable statements concerning only perceptible concrete symbols, such as the numerals 0, S0, SS0, SSS0. The purely formal component consists of ideal statements, statements that involve unbounded quantification over infinite domains such as the natural numbers. All such statements are strictly meaningless, according to Hilbert. Their introduction into mathematical theories was to be justified on purely instrumental grounds. They provide the mathematician with an extremely powerful, but in principle dispensable, means of proving facts about the real finitary subject matter of mathematics.

Hilbert's program was to show, using only finitary methods, that the introduction of such ideal statements into arithmetic could never lead to any false finitary statement becoming derivable. This is equivalent to proving using only finitary methods that classical arithmetic is consistent. He hoped to establish the same result for set theory, thereby establishing that the threat of inconsistency implied by the paradoxes could be guaranteed not to arise there either. "No one," wrote Hilbert, "shall drive us out from the paradise that Cantor has created for us" (Benacerraf and Putnam [1983], p. 191).

There is a fairly broad consensus that Kurt Gödel's second incompleteness theorem shows that Hilbert's program is unachievable, even at the level of arithmetic. Let T be any standard formal system for arithmetic. Suppose there was a finitary consistency proof for T. Then that proof could be formalized as a derivation in T of a formula expressing the consistency of T. However, by Gödel's second incompleteness theorem, no consistent formal system for arithmetic can contain such a derivation. It follows that the goal of Hilbert's program is unachievable for arithmetic and so also for set theory.

Intuitionism

A different response to the paradoxes, current at the time Hilbert was writing, was the intuitionist account of mathematics, proposed by the mathematician Luitzen Egbertus Jan Brouwer and developed by Arend Heyting. Intuitionism can be thought of as denying the first premise of the argument for platonism by claiming that although mathematics does constitute a body of statements that can be true or false, the truth or falsity of a mathematical statement is not independent of human beings.

The platonist thinks of the natural numbers as an infinite domain of objects that exist independently of human thought and that make arithmetical statements objectively true or false. By contrast, intuitionists such as Brouwer and Heyting think of the natural numbers as mental constructions, objects that are created by the human mind. On this view, what makes a mathematical statement true or false is not the existence of objects that are independent of human beings, but the existence of a certain kind of mental construction, a proof (though not a proof in a formal system).

This conception of mathematical truth led the intuitionists to reject the law of excluded middle, as applied to mathematics. That is, they denied the universal validity of the logical schema "Either A or not-A." For on the intuitionist view, a mathematical conjecture for which neither proof nor disproof has yet been constructed, cannot be said to be either true or false.

Although paradox may be avoided in the intuitionistic reconstruction of mathematics, many contemporary philosophers would reject it. One reason is the apparent truncation of classical mathematics necessitated by intuitionism; many theorems of classical analysis and set theory are false when interpreted intuitionistically. A deeper reason may be a distrust of the reforming nature of the intuitionism. The role of philosophy, it is thought, should be to provide an account or interpretation of mathematics as it actually is, not to reformulate or remake mathematics in a new image.

An exception to this general trend is Michael Dummett. A widely accepted philosophical thesis has it that the meaning of a statement is given by its truth conditions. Dummett argues that this is empty, unless accompanied by a substantive account of truth; an account which goes beyond the mere equivalence of 'P is true' with P. But meaning cannot be explained in terms of a concept of truth according to which truth is something that may apply to a statement quite independently of whether it is possible to know that it does, for then our knowledge of the meaning of a statement would not always be capable of being made manifest by publicly observable behaviour—a condition which is necessary for meaning to be communicable. Instead, meaning must be explained in terms of verification conditions; to know the meaning of a mathematical statement is to know what would count as a proof of it. Thus the argument leads to a version of intuitionism; to say that a mathematical statement is true is to say that we have a proof, while to say that it is false is to say that we have a disproof.

Dummett's argument depends only on very general considerations concerning the communicability of meaning. If valid, the argument would apply to statements of any kind whatsoever and not just to mathematical ones. For example, it would be a consequence that perfectly ordinary statements about the past which can no longer be verified or refuted could not be considered either true or false, unless it could be shown that there is some special feature of our use of such statements which makes a verification transcendent account of their meaning possible.

The Indispensability Argument

The epistemological objection to platonism is one aspect of a more general problem for empiricism. Mathematics appears to be highly non-empirical, in both its subject-matter and its methodology. Empirical evidence does not appear relevant to mathematics. However if, as the empiricist asserts, all our knowledge is ultimately empirical, there seems to be no good reason for thinking that mathematics is true at all. The logical positivist's claim that mathematics is analytic, or true only in virtue of meaning, was an attempt to solve this problem by showing how mathematical statements could be true, though independent of all empirical evidence.

In a now classic series of papers written in the late 1940s and early 1950s, Quine launched a major critique of this conventionalist solution to the problem, while also developing a significant alternative account of the structure of empirical knowledge and the place of mathematics within that structure. Quine argued that the mistake made by earlier empiricists was to think that individual statements can be tested empirically in isolation from each other. Instead, it must be recognized that our scientific beliefs form an interlocked system or web that "faces the tribunal of experience as a corporate body." (Quine, 'Two Dogmas of Empiricism', in 'From a logical point of view', p. 41). Mathematical beliefs form an indispensable part of this system and are therefore justified to the extent that they contribute to the goals of scientific prediction and explanation. In Quine's view the mathematics used in a successfully confirmed scientific theory is confirmed along with the rest of that theory. Mathematical objects, like numbers and sets, are theoretical posits, epistemologically on a par with electrons and photons.

Quine draws a further conclusion. If mathematics can be supported by empirical evidence, it can also be undermined by it. Our mathematical beliefs are open to empirical falsification and revision, in just the same way as our scientific beliefs. The illusion of a difference between mathematical and other scientific statements is generated, according to Quine, by pragmatic considerations. We are far more reluctant to revise the mathematical and logical components of our scientific theories because these are so deeply embedded in the system of total science that altering them would result in a major restructuring of the entire system. But if the result of such a restructuring was an overall simplification or improvement in the total system of science, then it would be perfectly rationally justified.

Quine's epistemology is significant because it provides a solution, consistent with empiricism, to the epistemological problem for platonism. Quine can accept all the premises in the argument for platonism given earlier. Mathematical statements can be taken to refer, as they appear to refer, to abstract objects such as numbers and sets. But the epistemological problem is resolved. Numbers and sets cannot be perceived, either directly or indirectly, but their utility in enabling us to predict and explain the world provides us with all the justification for their existence we need or could ever be entitled to.

The indispensability of mathematics in science has two aspects: one emphasized by Quine, the other by Hilary Putnam. Quine argues from the indispensability of mathematics in the derivation of the observation statements that confirm or disconfirm scientific theories and hypotheses. Putnam emphases a different aspect of the indispensability of mathematics in science. Mathematics is used in science, not only in deriving predictions from theories but also in formulating the empirical hypotheses of those theories. Consider Boyle's law, for example, which states the relationship between the pressure, temperature, and volume of a fixed quantity of gas enclosed in some container. The law states that the pressure of the gas is equal to a constant multiplied by the temperature of the gas, divided by the volume.
P = kT / V
Pressure, temperature, and volume are all numerical quantities. The pressure of the gas in kilopascals at a certain time is a real number, as are the volume in cubic centimeters and the temperature in degrees Celsius at a time. The law states that a certain mathematical relationship always holds between these real numbers. Boyle's law is therefore just as much committed to the existence of real numbers and functions as it is to the existence of gases. In this way, realism about physical theory leads to realism about mathematical objects such as numbers.

The Quine-Putnam argument allows for the empirical justification not only of the often highly specific mathematical statements (such as numerical equations) used to derive predictions from a theory but also of any more general mathematical statements, such as set-theoretic axioms, which imply them. Boyle's law can be derived from the more fundamental laws of thermodynamics. Hence any empirical confirmation of Boyle's law accrues also to the thermodynamic laws used to derive it. In just the same way, since arithmetic can be reduced to set theory, the numerical equations used to derive predictions from Boyle's law can be derived from the axioms of set theory. Hence any empirical confirmation of those equations accrues also to the axioms of set theory. In this way, it might be hoped that a great deal of even abstract mathematics can be justified by means of the Quine-Putnam argument.

See also Frege, Gottlob; Mathematics, Foundations of; Platonism and the Platonic Tradition; Quine, Willard Van Orman; Russell, Bertrand Arthur William.

Bibliography

On the historical development of the mathematics of number, see Morris Kline, Mathematical Thought from Ancient to Modern Times (New York: Oxford University Press, 1972).

For Euclid's Elements, the standard reference is T. L Heath, The Thirteen Books of Euclid's Elements (New York: Dover Publications, 1956).

Collections of mathematical papers by Frege, Cantor, Gödel, Zermelo, and others can be found in D. E. Smith, ed. A Source Book in Mathematics (New York: Dover Publications, 1959) and J. van Heijenoort, ed. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (Cambridge, MA: Harvard University Press, 1967).

For the philosophy of mathematics, the standard collection of papers is Paul Benacerraf and Hilary Putnam, eds. Philosophy of Mathematics: Selected Readings (2nd ed. New York: Cambridge University Press, 1983). Also useful is W. D. Hart, ed. The Philosophy of Mathematics (New York: Oxford University Press, 1996).

Frege's classic Die Grundlagen der Arithmetic (Breslau, Poland: Koebner, 1884) is translated by J. L Austin as The Foundations of Arithmetic (Oxford, U.K.: Basil Blackwell, 1953).

For philosophical expositions of Frege's work, see Michael Dummett, Frege Philosophy of Mathematics (London: Duckworth, 1991) and Crispin Wright, Frege's Conception of Numbers as Objects (Aberdeen, Scotland: Aberdeen University Press, 1983). W. Demopoulos, ed. Frege's Philosophy of Mathematics (Cambridge, MA: Harvard University Press, 1995) provides an excellent collection of papers on the technical aspects of Frege's work.

On the epistemological problem for platonism, see Paul Benacerraf, "Mathematical Truth," Journal of Philosophy 70 (1973): 661–679.

For a survey of work on physicalist accounts of mathematical, see A. D. Irvine, ed. Physicalism in Mathematics. Dordrecht, Netherlands: Kluwer Academic, 1990.

On structuralism and the ontological problem for platonism, see Paul Benacerraf, "What Numbers Could Not Be," Philosophical Review 74 (1965): 47–73; Hilary Putnam, "Mathematics without Foundations," Journal of Philosophy 64 (1965): 5–22; and Charles Parsons, "The Structuralist View of Mathematical Objects," Synthese 90 (1990): 303–346.

On the conventionalist account of mathematics and logic, see A. J. Ayer, Language, Truth, and Logic (London: V. Gollancz, 1946); Rudolf Carnap, "Empiricism, Semantics, and Ontology" (1956) and Carl Gustav Hempel, "On the Nature of Mathematical Truth," American Mathematical Monthly 52 (1954), both reprinted in Paul Benacerraf and Hilary Putnam, eds. Philosophy of Mathematics: Selected Readings (2nd ed. New York: Cambridge University Press, 1983).

Quine's critiques of conventionalism can be found in "Truth by Convention" and "Carnap and Logical Truth" in Quine, Ways of Paradox and Other Essays (Cambridge, MA: Harvard University Press, 1976).

For the indispensability argument, see Quine, "Two Dogmas of Empiricism" and "On What There Is" in his From a Logical Point of View (2nd ed. Cambridge, MA: Harvard University Press, 1980). See also Quine, Word and Object (Cambridge, MA: MIT Press, 1960).

Putnam's exposition of Quine's argument is to be found in Hilary Putnam, Philosophy of Logic (London: Allen and Unwin, 1971).

On Field's program, see Hartry Field, Science without Numbers: A Defence of Nominalism (Princeton, NJ: Princeton University Press, 1980). For a critical analysis, see Stewart Shapiro, "Conservativeness and Incompleteness," Journal of Philosophy 80 (1983): 521–531. A. D. Irvine, ed. Physicalism in Mathematics (Dordrecht, Netherlands: Kluwer Academic, 1990) also contains a number of relevant articles on nominalism.

David Hilbert's formalist program is outlined in a readable way in his "On the Infinite" (1926), which is reprinted in Paul Benacerraf and Hilary Putnam, eds. Philosophy of Mathematics: Selected Readings (2nd ed. New York: Cambridge University Press, 1983, pp. 83–201). A scholarly exposition of this and other foundational programs can be found in Marcus Giaquinto, The Search for Certainty: A Philosophical Account of Foundations of Mathematics (Oxford, U.K.: Clarendon Press, 2002).

On intuitionism, see the papers by Brouwer and Heyting in Paul Benacerraf and Hilary Putnam, eds. Philosophy of Mathematics: Selected Readings (2nd ed. New York: Cambridge University Press, 1982). Michael Dummett's Elements of Intuitionism (Oxford, U.K.: Clarendon Press, 1977) is a standard reference work on intuitionistic mathematics and philosophy. Dummett's own argument for an intuitionistic interpretation of mathematics can be found in "The Philosophical Basis of Intuitionistic Logic" in his Truth and Other Enigmas (London: Duckworth, 1975).

John Bigelow (2005)

Sam Butchart (2005)

num·ber / ˈnəmbər/ • n. 1. an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations and for showing order in a series or for identification: she dialed the number carefully | an even number. ∎  (numbers) dated arithmetic: the boy was adept at numbers.2. a quantity or amount: the company is seeking to increase the number of women on its staff | the exhibition attracted vast numbers of visitors. ∎  (a number of) several: we have discussed the matter on a number of occasions. ∎  a group or company of people: there were some distinguished names among our number. ∎  (numbers) a large quantity or amount, often in contrast to a smaller one; numerical preponderance: the weight of numbers turned the battle against them.3. a single issue of a magazine: the October number of “Travel.” ∎  a song, dance, piece of music, etc., esp. one of several in a performance: they go from one melodious number to another. ∎  inf. a thing, typically an item of clothing, of a particular type, regarded with approval or admiration: Yvonne was wearing a little black number.4. Gram. a distinction of word form denoting reference to one person or thing or to more than one. See also singular (sense 2), plural, count noun, and mass noun. ∎  a particular form so classified.• v. [tr.] 1. amount to (a specified figure or quantity); comprise: the demonstrators numbered more than 5,000. ∎  include or classify as a member of a group: the orchestra numbers Brahms among its past conductors.2. (often be numbered) mark with a number or assign a number to, typically to indicate position in a series: each document was numbered consecutively. ∎  count: strategies like ours can be numbered on the fingers of one hand. ∎  assess or estimate the size or quantity of (something) to be a specified figure: he numbers the fleet at a thousand.PHRASES: any number of any particular whole quantity of: the game can involve any number of players. ∎  a large and unlimited quantity or amount of: the results can be read any number of ways.by numbers following simple instructions identified by numbers or as if identified: painting by numbers.by the numbers following standard operating procedure. ∎  all together with a shouted-out count.someone's/something's days are numbered someone or something will not survive or remain in a position of power or advantage for much longer: my days as director were numbered.do a number on inf. treat someone badly, typically by deceiving, humiliating, or criticizing them in a calculated and thorough way.have someone's number inf. understand a person's real motives or character and thereby gain some advantage.have someone's number on it inf. (of a bomb, bullet, or other missile) destined to find a specified person as its target.someone's number is up inf. the time has come when someone is doomed to die or suffer some other disaster or setback.without number too many to count: they forgot the message times without number.

NUMBER1 A concept associated with quantity, size, measurement, etc., and represented by a word such as three, a symbol such as 3, a group of words such as eighty-three point five, or a group of symbols such as 83.5. Every number, regardless of the language in which it is expressed, occupies a unique position in a series, such as 3 in the series 1, 2, 3, 4, 5, …, enabling it to be used in such arithmetical processes as addition, subtraction, multiplication, and division. There are two basic kinds of number in such languages as English and French: cardinal numbers (the term deriving ultimately from Latin cardo/cardinis a hinge: that is, something on which other things turn or depend), denoting quantity and not order (as in 1, 2, 3, 4); and ordinal numbers (the term deriving ultimately from Latin ordo/ordinis order), denoting relative position in a sequence (as in 1st, 2nd, 3rd, 4th). Grammatically, the number system of a language contrasts with its system of quantifiers: for example, one house with a house, and two/three/forty people, etc., with some people, several people, and many people, etc.

Numbers as words

A spoken number is a WORD or phrase in a language, but a written number may be realized as either a word or phrase or a symbol or groups of symbols, usually a figure such as 1, 2, 12, 21. Written words are generally used for low numbers, from one to ten or twelve (as in the phrases three blind mice, the seven wonders of the world, and the twelve signs of the Zodiac). They are also often used for numbers up to 100 (with hyphenation for compound forms such as twenty-one and eighty-three) and for large round figures as in a thousand years and four million visitors a year. Words may or may not be used to express percentages, which may be given as ten per cent, 10 per cent, or 10% depending on house style or personal preference. Most house styles and editors aim for consistency in whichever forms they have chosen.

Numbers as symbols

Arabic figures are commonly used for numbers above ten or twelve (as in The ship sank with the loss of 18 lives), before abbreviations (as in 8 pm for eight o'clock in the evening, 7K for seven thousand, and 3m for three million), and for dates, addresses, and exact sums of money. Large numbers such as 118,985 are usually given as figures; when spoken, there is one significant difference between British and American usage: BrE always has and after hundred, as in one hundred and eighteen thousand, nine hundred and eighty-five, while AmE generally does not, as in one hundred eighteen thousand, nine hundred eighty-five. In large numbers, commas are generally used after the figures representing millions and thousands (1,345,905), but spaces are also, perhaps increasingly, used for this purpose (1 345 905); commas or spaces may or may not be used for thousands alone (2,345 and 2 345), for which solid numbers are also common (2345). Telephone numbers are generally written with spaces between regional and local numbers (01223 245999), and reference numbers are generally solid (N707096). Plural s after a set of numbers is often preceded by an apostrophe, as in 3's and 4's or the 1980's, but many house styles and individuals now favour 3s and 4s and the 1980s.

Numbers in -illion

Formerly, BrE and AmE differed greatly in their use of numbers representing multiples of million: for example, in Britain, France, and Germany, billion was ‘one million million’, or 10¹² (10 to the power 12), while in the US and Canada it was ‘one thousand million’ or 10⁹. The North American equivalent to the British billion was the trillion. In the last decades of the 20c, however, the North American use has become universal, providing the set million, billion, trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion. The -illion pattern has prompted some word-play, especially in AmE, that makes use of various initial consonants and syllables: ‘The savings-and-loan industry bailout, which as of yesterday afternoon was expected to cost taxpayers $752.6 trillion skillion, is now expected to cost $964.3 hillion jillion bazillion’ ( Dave Barry, ‘Give or Take a Whomptillion’, International Herald Tribune, 13 June 1990). The widely-used zillion, with its end-of-alphabet prefix, usually suggests the ultimate in facetious scale, but Barry's ba-adds even more force. See LETTER 1, QUANTIFIER.

NUMBER2. A grammatical category used in describing parts of speech that show contrasts of PLURAL, SINGULAR, dual, etc. In English, the number system is basically a two-term contrast of singular and plural, shown in nouns and some pronouns and determiners, and to some extent in verbs. Even dual words, such as both, either, neither, take singular or plural verb CONCORD: both taking the plural; either, neither usually taking the singular. English nouns, as far as number is concerned, can be divided into: singular only, plural only, and words that can be both. Singular-only nouns are: (1) Uncountable nouns which can occur with such uncountable-specific words as much, little: much money, little sugar. (2) Most proper nouns: Edinburgh, the Thames (in which other restrictions apply). Plural only nouns are: (1) Countable: people in six people, but not in the European peoples. (2) Usually uncountable: not enough clothes (not *six clothes); many thanks (not *five thanks); trousers (a pair of trousers and not usually three trousers). The vast majority of countable nouns can be both singular and plural (book/books, fox/foxes, mouse/mice), but a few have no distinct plural form (as with one sheep/three sheep). Many nouns, however, have both countable and uncountable uses, in which case they may have a plural in some uses (What an excellent wine/What excellent wines!) but not in others (I never drink wine). Pronouns having distinct singular and plural forms include personal, reflexive, and possessive. Number contrast is neutralized with you, but the second-person reflexive forms distinguish yourself and yourselves. Demonstrative pronouns also have separate forms, singular this, that being used both with singular countable nouns (this restaurant) and with uncountable nouns (this food). Number contrast in verbs, except in the verb be, is confined to the distinct third-person singular tense form (look/looks). See PRONOUN, QUANTIFIER.

number an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations and for showing order in a series or for identification.
the number of the beast the number 666, numerologically representing the name of the Antichrist of the Revelation 13:18, ‘Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is Six hundred threescore and six.’ It has been suggested that the number given could be a coded reference to Nero: the numerical value of Nero Caesar, written in Hebrew letters, adds up to 666.
Number Ten a name for 10 Downing Street, the official London home of the British Prime Minister.

See also back number, golden number, have one's name and number on it, numbers, prime number.