In English and other natural languages there occur also *common names* (common nouns), such a common name being thought of as if it could serve as a name of anything belonging to a specified class or having specified characteristics. Under usual translations into symbolic notation, common names are replaced by proper names of classes or of class concepts; and this would seem to provide the best logical analysis. In actual English usage, however, a common noun is often more nearly like a *variable* (q. v.) having a specified range. -- A.C.

Where a particular (Interpreted) system does not contain symbols for formulas, it may be desirable to employ Gödel's device for associating (positive integral) numbers with formulas, and to consider the relation between a number and that which the associated formula denotes. This we shall call the *numerical name relation* and distinguish it from the relation between a formula and that which it denotes by calling the latter the *semantical name relation*.

In many (interpreted) logistic systems -- including such as contain, with their usual interpretations, the Zermelo set theory, or the simple theory of types with axiom of infinity, or the functional calculus of second order with addition of Peano's postulates for arithmetic -- it is impossible without contradiction to introduce the numerical name relation with its natural properties, because Grelling's paradox or similar paradoxes would result (see *paradoxes, logical*). The same can be said of the semantical name relation in cases where symbols for formulas are present.

Such systems may, however, contain partial name relations which function as name relations in the case of some but not all of the formulas of the system (or of their associated Gödel numbers).

In particular, it is normally possible -- at least it does not obviously lead to contradiction in the case of such systems as the Zermelo set theory or the simple theory of types (functional calculus of order omega) with axiom of infinity -- to extend a system L_{1} into a system L_{2} (the semantics of L_{1} in the sense of Tarski), so that L_{2} shall contain symbols for the formulas of L_{1}, and for the essential syntactical relations between formulas of L_{1}, and for a relation which functions as a name relation as regards all the formulas of L_{1} (or, in the case of the theory of types, one such relation for each type), together with appropriate new primitive formulas. Then L_{2} may be similarly extended into L_{3}, and so on through a hierarchy of systems each including the preceding one as a part.

Or, if L_{1} contains symbols for positive integers, we may extend L_{1} into L_{2} by merely adding a symbol for a relation which functions as a numerical name relation as regards all numbers of formulas of L_{1} (or one such relation for each type) together with appropriate new primitive formulas; and so on through a hierarchy of systems L_{1}, L_{2}, L_{3}, . . . .

See further *Semantics*; *Semtotic* 2; *Truth, Semantical*. -- A.C.

The general philosophical position which has as its fundamental tenet the proposition that the natural world is the whole of reality. "Nature" and "natural world" are certainly ambiguous terms, but this much is clear in thus restricting reality, naturalism means to assert that there is but one system or level of reality, that this system is the totality of objects and events in space and time; and that the behavior of this system is determined only by its own character and is reducible to a set of causal laws. Nature is thus conceived as self-contained and self-dependent, and from this view spring certain negations that define to a great extent the influence of naturalism. First, it is denied that nature is derived from or dependent upon any transcendent, supernatural entities. From this follows the denial that the order of natural events can be intruded upon. And this in turn entails the denial of freedom, purpose, and transcendent destiny.

Within the context of these views there is evidently allowance for divergent doctrines, but certain general tendencies can be noticed. The metaphysics of naturalism is always monistic and if any teleological element is introduced it is emergent. Man is viewed as coordinate with other parts of nature, and naturalistic psychology emphasizes the physical basis of human behavior; ideas and ideals are largely treated as artifacts, though there is disagreement as to the validity to be assigned them. The axiology of naturalism can seek its values only within the context of human character and experience, and must ground these values on individual self-realization or social utility; though again there is disagreement as to both the content and the final validity of the values there discovered. Naturalistic epistemologies have varied between the extremes of rationalism and positivism, but they consistently limit knowledge to natural events and the relationships holding between them, and so direct inquiry to a description and systematization of what happens in nature. The beneficent task that naturalism recurrently performs is that of recalling attention from a blind absorption in theory to a fresh consideration of the facts and values exhibited in nature and life.

*In aesthetics*: The general doctrine that the proper study of art is nature. In this broad sense, artistic naturalism is simply the thesis that the artist's sole concern and function should be to observe closely and report clearly the character and behavior of his physical environment. Similarly to philosophical naturalism, aesthetic naturalism derives much of its importance from its denials and from the manner in which it consequently restricts and directs art. The artist should not seek any "hidden" reality or essence; he should not attempt to correct or complete nature by either idealizing or generalizing; he should not impose value judgments upon nature; and he should not concern himself with the selection of "beautiful" subjects that will yield "aesthetic pleasure". He is simply to dissect and describe what he finds around him. Here, it is important to notice explicitly a distinction between naturalism and romanticism (q.v.): romanticism emphasizes the felt quality of things, and the romanticist is primarily interested in the experiences that nature will yield, naturalism emphasizes the objective character of things, and is interested in nature as an independent entity. Thus, romanticism stresses the intervention of the artist upon nature, while naturalism seeks to reduce this to a minimum.

Specifically, naturalism usually refers to the doctrines and practices of the 19th century school of realism which arose as the literary analogue of positivism, and whose great masters were Flaubert, Zola, and de Maupassant. The fundamental dogma of the movement, as expressed by Zola in "Le Roman experimental" and "Les Romanciers naturalistes", states that naturalism is "the scientific mdhod applied to literature". Zola maintains that the task of the artist is to report and explain what happens in nature, art must aim at a literal transcript of reality, and the artist attains this by making an analytic study of character, motives, and behavior. Naturalism argues that all judgments of good and bad are conventional, with no real basis in nature, so art should seek to understand, not to approve or condemn. Human behavior is regarded as largely a function of environment and circumstances, and the novelist should exhibit these in detail, with no false idealizing of character, no glossing over of the ugly, and no appeal to supposed hidden forces. -- I.J.

- The objective as opposed to the subjective.
- An objective standard for values as opposed to custom, law, convention.
- The general cosmic order, usually conceived as divinely ordained, in contrast to human deviations from this.
- That which exists apart from and uninfluenced by man, in contrast with art.
- The instinctive or spontaneous behavior of man as opposed to the intellective.

In *Aristotle's philosophy*: (1) the internal source of change or rest in an object as such, in distinction from art, which is an external source of change. Natural beings are those that have such an internal source of change. Though both matter and form are involved in the changes of a natural being, its nature is ordinarily identified with the form, as the active and intelligible factor. (2) The sum total of all natural beings. See *Aristotelianism*. -- G.R.M.

A proposition may also be said to be necessary if it is a consequence of some accepted set of propositions (indicated by the context), even if this accepted set of propositions is not held to be *a priori*. See *Necessity*.

That a propositional function F is necessary may mean simply (x)F(x), or it may mean that (x)F(x) is necessary in one of the preceding senses. -- A.C.

- logical or mathematical necessity,
- physical necessity, and
- moral necessity.

Logical, physical, and moral necessity are founded in logical, physical, and moral laws respectively. Anything is logically necessary the denial of which would violate a law of logic. Thus in ordinary commutative algebra the implication from the postulates to ab-ba is logically necessary, since its denial would violate a logical law (viz. the commutative rule) of this system.

Similarly, physically necessary things are those whose denial would violate a physical or natural law. The orbits of the planets are said to be physically necessary. Circular orbits for the planets are logically possible, but not physically possible, so long as certain physical laws of motion remain true. Physical necessity is also referred to as "causal" necessity.

As moral laws differ widely from logical and physical laws, the type of necessity which they generate is considerably different from the two types previous defined. Moral necessity is illustrated in the necessity of an obligation. Fulfillment of the obligation is morally necessary in the sense that the failure to fulfill it would violate a moral law, where this law is regarded as embodying some recognized value. If it is admitted that values are relative to individuals and societies, then the laws embodying these values will be similarly relative, and likewise the type of thing which these laws will render morally necessary.

While these three types of necessity are generally recognized by philosophers, the weighting of the distinctions is a matter of considerable divergence of view. Those who hold that the distinctions are all radical, sharply distinguish between logical statements, statements of fact, and so-called ethical or value statements. On the other hand, the attempt to establish an a priori ethics may be regarded as an attempt to reduce moral necessity to logical necessity; while the attempt to derive ethical evaluations from the statements of science, e.g. from biology, is an attempt to reduce moral necessity to physical or causal necessity. -- F.L.W.

The negation of a proposition p is the proposition ∼p (see *Logic, formal*, § 1). The negation of a monadic propositional function F is the monadic propositional function λ[∼F(x)]; similarly for dyadic propositional functions, etc.

Or the word negation may be used in a syntactical sense, so that the negation of a sentence (formula) A is the sentence ∼A. -- A. C.

(a) English New Realists: Less radical in that mind was given a status of its own character although a part of its objective environment. Among distinguished representatives were: G. E. Moore, Bertrand Russell, S. Alexander, T. P. Nunn, A. Wolf, G. F. Stout,

(b) American New Realists: More radical in that mind tended to lose its *special* status in the order of things. In psychology this school moved toward behaviorism. In philosophy they were extreme pan-objectivists. Distinguished representatives: F. J. E. Woodbridge, G. S. Fullerton, E. B. McGilvary and six platformists (so-called because of their collaboration in a volume *The New Realism*, published 1912): E. B. Holt, W. T. Marvin, W. P. Montague, R. B. Perry, W. B. Pitkin, E. G. Spaulding. The American New Realists agreed on a general platform but differed greatly among themselves as to theories of reality and particular questions. -- V.F.

- We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.
- Therefore to the same natural effects we must, as far as possible, assign the same causes.
- The qualities of bodies, which admit neither intension nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.
- In experimental philosophy we are to look upon propositions collected by general induction from phaenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phaenomena occur, by which they may either be made more accurate, or liable to exceptions.

He was the first to recognize a fundamental critical difference between the philosopher and the scientist. He found those genuine ideals in the pre-Socratic period of Greek culture which he regarded as essential standards for the deepening of individuality and real culture in the deepest sense, towards which the special and natural sciences, and professional or academic philosophers failed to contribute. Nietzsche wanted the philosopher to be prophetic, originally forward-looking in the clarification of the problem of existence. Based on a comprehensive critique of the history of Western civilization, that the highest values in religion, morals and philosophy have begun to lose their power, his philosophy gradually assumed the will to power, self-aggrandizement, as the all-embracing principle in inorganic and organic nature, in the development of the mind, in the individual and in society. More interested in developing a philosophy of life than a system of academic philosophy, his view is that only that life is worth living which develops the strength and integrity to withstand the unavoidable sufferings and misfortunes of existence without flying into an imaginary world.

His major works are: *Thus Spake Zarathustra*; *Beyond Good and Evil*, and *Genealogy of Morals*. -- H.H.

- Nothing exists;
- Even if something did exist it could not be known;
- Even if it were known this knowledge could not be communicated.

In Greek philosophers The exercise of nous, or reason, the activity of intellectual apprehension and intuitive thought. See *Nous; Aristotelianism*. -- G.RM.

(Ger. noetisch) In Husserl: Of or pertaining to noesis. See note under *noesis*. -- D.C.

- Non-existence or the non-existent; absence or piivation of existence or the existent;
- absence of determinateness or what is thus indeterminate;
- unreality of the unreal -- either lack of any reality or what is so lacking (absence, negation, or privation of reality), or lack of a particular kind of reality or what is so lacking; otherness or existents of another order of reality than a specified type; failure to fulfill the defining criteria of some category, or what so fails;
- a category encompassing any of the above.

Attempts to prove the parallel postulate from the other postulates of Euclidean geometry were unsuccessful. The undertaking of Saccheri
(1733) to make a proof by *reductio ad absurdum* of the parallel postulate by deducing consequences of its negation did, however, lead to his developing many of the theorems of what is now known as hyperbolic geometry. The proposal that this hyperbolic geometry, in which Euclid's parallel postulate is replaced by its negation, is a system equally valid with the Euclidean originated with Bolyai and Lobachevsky (independently, c 1825). Proof of the self-consistency of hyperbolic geometry, and thus of the impossibility of Saccheri's undertaking, is contained in results of Cayley (1859) and was made explicit by Klein in 1871; for the two-dimensional case another proof was given by Beltrami in 1868.

The name *non-Euclidean geometry* is applied to hyperbolic geometry and generally to any system in which one or more postulates of Euclidean geometry are replaced by contrary assumptions. (But geometries of more than three dimensions, if they otherwise follow the postulates of Euclid, are not ordinarily called non-Euclidean.)

Closely related to the hyperbolic geometry is the elliptic geometry, which was introduced by Klein on the basis of ideas of Riemann. In this geometry lines are of finite total length and closed, and every two coplanar lines intersect in a unique point.

Still other non-Euclidean geometries are given an actual application to physical space -- or rather, space-time -- in the General Theory of Relativity.

Contemporary ideas concerning the abstract nature of *mathematics* (q. v.) and the status of applied geometry have important historical roots in the discovery of non-Euclidean geometries. -- A.C.

- G. Saccheri,
*Euclides Vindicatus*, translated into English by G. B Halsted, Chicago and London, 1920.- H. P. Manning,
*Non-Euclidean Geometry*, 1901.- J. L. Coolidge,
*The Elements of Non-Euclidean Geometry*, Oxford. 1909.

- General: Standard for measure. Pattern. Type.
- In ethics: Standard for proper conduct. Rule for right action.
- In axiology: Standard for judging value or evaluation.
- In aesthetics: Standard for judging beauty or art. Basis for criticism,
- In logic: Rule for valid inference.
- In psychology: Class average test score.

PROPOSITIONAL CALCULUS (see *Logic, formal*, § 1, and *strict implication*)

pq, the conjunction of p and q, "p and q." Instead of simple juxtaposition of the propositional symbols, a dot is sometimes written between, as p·q. Or the common abbreviation for *and* may be employed as a logical symbol, p & q.
Or an inverted letter ∨, usually from a gothic font, may be used. In the Lukasiewicz notation for the propositional calculus, which avoids necessity for parentheses, the conjunction of p and q id Kpq.

p ∨ q, the inclusive disjunction of p and q, "p or q." Frequently the letter ∨ is from a gothic font. In the Lukasiewicz notation, Apq is employed.

∼p, the negation of p, "not p." Instead of ∼, a dash − may be used, written either before the propositional symbol or above it. Heyting adds a short downward stroke at the right end of the dash (a notation which has come to be associated particularly with the intuitionistic propositional calculus and the intuitionistic concept of negation). Also employed is an accent after the propositional symbol (but this more usual as a notation for the complement of a class). In the Lukasiewicz notation, the negation of p is Np.

p ⊃ q, the material implication of q by p, "if p then q." Also employed is a horizontal arrow, p → q. The Lukasiewicz notation is Cpq.

p ≡ q, the material equivalence of p and q, "p if and only if q." Another notation which has sometimes been employed is p ⊃⊂ q. Other notations are a double horizontal arrow, with point at both ends, and two horizontal arrows, one above the other, one pointing forward and the other back. The Lukasiewicz notation is Epq.

p + q, the exclusive disjunction of p and q, "p or q but not both." Also sometimes used is the sign of material equivalence ≡ with a vertical or slanting line across it (non-equivalence). In connection with the Lukasiewicz notation, Rpq has been employed.

p|q, the alternative denial of p and q, "not both p and q." -- For the dual connective, joint denial ("neither p nor q"), a downward arrow has been used.

p 3 q, the strict implication of q by p, "p strictly implies q."

p = q, the strict equivalence of p and q, "p strictly implies q and q strictly implies p." Some recent writers employ, for strict equivalence, instead of Lewis's =, a sign similar to the sign of material equivalence, ≡, but with four lines instead of three.

Mp, "p is possible." This is Lukasiewicz's notation and has been used especially in connection with his three-valued propositional calculus. For the different notion of possibility which is appropriate to the calculus of strict implication, Lewis employs a diamond.

CLASSES (see *class*, and *logic, formal*, §§ 7, 9):

x∈a, "x is a member of the class a," or, "x is an a." For the negation of this, sometimes a vertical line across the letter epsilon is employed, or a ∼ above it.

a ⊂ b, the inclusion of the class a in the class b, "a is a subclass of b." This notation is usually employed in such a way that a ⊂ b does not exclude the possibility that a = b. Sometimes, however, the usage is that a ⊂ b ("a is a proper subclass of b") does exclude that a = b; and in that case another notation is used when it is not meant that a = b is excluded, the sign = being either surcharged upon the sign ⊂ or written below it (or a single horizontal line below the ⊂ may take the place of =).

∃!a, "the class a is not empty [has at least one member]," or, "a's exist."

ιx, or ι‘x, or [x] -- the unit class of x, i.e., the class whose single member is x.

V, the universal class. Where the algebra of classes is treated in isolation, the digit 1 is often used for the universal class.

Λ, the null or empty class. Where the algebra of classes is treated in isolation, the digit 0 is often used.

−a, the complement of a, or class of non-members of the class a. An alternative notation is a a'.

a ∪ b, the logical sum, or union, of the classes a and b. Alternative notation, a + b.

a ∩ b, the logical product, or intersection, or common part, of the classes a and b. Alternative notation, ab.

RELATIONS (see *Relation*, and *Logic, formal*, § 8; (where a notation used in connection with relations is here given as identical with a corresponding notation for classes, the relational notation will also often be found with a dot added to distinguish it from the one for classes):

xRy, "x has [or stands in, or bears] the relation R to y."

R ⊂ S, "the relation R is contained in [implies] the relation S."

&esixt;!R, "the relation R is not null [holds in at least one instance]."

A downward arrow placed between (e.g.) x and y denotes the relation which holds between x and y (in that order) and in no other case.

V, the universal relation. Schröder uses 1.

Λ, the null relation. Schröder uses 0.

−R, the contrary, or negation, of the relation R. The dash may also be placed over the letter R (or other symbol denoting a relation) instead of before it.

R ∪ S, the logical sum of the relations R and S, "R or S." Schröder uses R+S.

R ∩ S, the logical product of the relations R and S, "R and S." Schröder uses R·S.

I, the relation of identity -- so that xIy is the same as x = y. Schröder uses 1'.

J, the relation of diversity -- so that xJy is the same as x ≠ y. Schröder uses 0'.

A breve ^{∪} is placed over the symbol for a relation to denote the converse relation. An alternative notation for the converse of R is Cnv‘R.

R + S, the relative sum of R and S. Schröder adds a leftward hook at the bottom of the vertical line in the sign +.

R|S, the relative product of R and S. Schröder uses a semicolon to symbolize the relative product, but the vertical bar, or sometimes a slanted bar, is now the usual notation.

R^{2}, the square of the relation R, i.e., R|R.

Similarly for higher powers of a relation, as R^{3}, etc.

R‘y, the (unique) x such that xRy, "the R of y." Frequently the inverted comma is of a bold square (bold gothic) style.

R‘‘b, the class of x's which bear the relation R to at least one member of the class b, "the R's of the b's." Then R‘‘ιy, or R‘‘ι‘y, is the class of x's such that xRy, "the R's of y."

A forward pointing arrow is placed over (e. g.) R to denote the relation of R‘&slquo;ιy to y. Similarly a backward pointing arrow placed over R denotes the relation of *the class of y's such that* xRy to x.

An upward arrow placed between (e.g.) a and b denotes the relation which holds between x and y¦ if and only if x∈a and y∈b.

The left half of an upward arrow placed between (e.g.) a and R denotes the relation which holds between x and y if and only if x∈a and xRy, in other words, the relation R with its domain limited to the class a.

The right half of an upward arrow placed between (e.g.) R and b denotes the relation which holds between x and y if and only if xRy and y∈b; in other words the relation R with its converse domain limited to b.

The right half of a double -- upward and downward -- arrow placed between (e.g.) R and a denotes the relation which holds between x and y if and only if xRy and both x and y are members of the class a; in other words, the relation R with its field limited to a.

D‘R, the domain of R.

(|‘R, the converse domain of R.

C‘R, the field of R.

R_{po}, the proper ancestral of R -- i.e., the relation which holds between x and y if and only if x bears the first or some higher power of the relation R to y (where the first power of R is R).

R* the ancestral of R -- i.e., the relation which holds between x and y if and only if x bears the zero or some higher power of the relation R to y (where the zero power of R is taken to be, either I, or I with its field limited to the field of R).

QUANTIFIERS (see *Quantifiers*, and *Logic, formal*, §§3, 6)

(x), universal quantification with respect to x -- so that (x)M may be read "for every x, M." An alternative notation occasionally met with, instead of (x), is (∀x), usually with the inverted A from a gothic or other special font. Another notation is composed of a Greek capital pi with the x placed either after it, or before it, or as a subscript. -- Negation of the universal quantifier is sometimes expressed by means of a dash, or horizontal line, over it.

(Ex), existential quantification with respect to x -- so that (Ex)M may be read "there exists an x such that M." The E which forms part of the notation may also be inverted; and, whether inverted or not, the E is frequently taken from a gothic or other special font. An alternative notation employs a Greek capital sigma with x either after it or as a subscript. -- Negation of the existential quantifier is sometimes expressed by means of a dash over it.

⊃_{x}, formal implication with respect to x. See definition in the article *logic, formal*, § 3.

≡_{x}, formal equivalence with respect to x. See definition in *logic, formal*, § 3.

∧_{x}. See definition in *logic, formal*, § 3.

⊃_{xy} or ⊃_{x,y} -- formal implication with respect to x and y. Similarly for formal implication with respect to three or more variables.

≡_{xy}, or ≡_{x,y} -- formal equivalence with respect to x and y. Similarly for formal equivalence with respect to three or more variables.

ABSTRACTION, DESCRIPTIONS (see articles of those titles):

λx, functional abstraction with respect to x -- so that λxM may be read "the (monadic) function whose value for the argument x is M."

x, class abstraction with respect to x -- so that x M may be read "the class of x's such that M." An alternative notation, instead of x , is x3.

xy, relation abstraction with respect to x and y -- so that xyM may be read "the relation which holds between x and y if and only if M."

(i x), description with respect to x -- so that (i x)M may be read "the x such that M."

E! is employed in connection with descriptions to denote existence, so that E!(ix)M may be read "there exists a unique x such that M."

OTHER NOTATIONS:

F(x), the result of application of the (monadic, propositional or other) function F to the argument x -- the value of the function F for the argument x -- ftF of x." Sometimes the parentheses are omitted, so that the notation is Fx. -- See the articles function, and propostttonal function.

F(x, y), the result of application of the (dyadic) function F to the arguments x and y. Similarly for larger numbers of arguments.

x = y, the identity or equality of x and y, "x equals y." See *logic, formal*, §§ 3, 6, 9.

x ≠ y, negation of x = y.

|- is the assertion sign. See *assertion, logical*.

Dots (frequently printed as bold, or bold square, dots) are used in the punctuation of logical formulas, to avoid or replace parentheses. There are varying conventions for this purpose.

→ is used to express definitions, the definiendum being placed to the left and the definiens to the right. An alternative notation is the sign = (or, in connection with the propositional calculus, ≡) with the letters Df, or df, written above it, or as a subscript, or separately after the definiens.

Quotation marks, usually single quotes, are employed as a means of distinguishing the name of a symbol or formula from the symbol or formula itself (see *syntax, logical*). A symbol or formula between quotation marks is employed as a name of that particular symbol or formula. E.g., 'p' is a name of the sixteenth letter of the English alphabet in small italic type.

The reader will observe that this use of quotation marks has not been followed in the present article, and in fact that there are frequent inaccuracies from the point of view of strict preservation of the distinction between a symbol and its name. These inaccuracies are of too involved a character to be removed by merely supplying quotation marks at appropriate places. But it is thought that there is no point at which real doubt will arise as to the meaning intended. -- Alonzo Church

In translation into logical notation, the word nothing is usually to be represented by the negation of an existential quantifier. Thus "nothing has the property F" becomes "∼(Ex)F(x)." -- A.C.

First are the *non-negative integers* 0, 1, 2, 3, . . . , for which the operations of addition and multiplication are determined. They are ordered by a relation not greater than -- which we shall denote by R -- so that, e.g., 0R0, 0R3, 2R3, 3R3, 57R218, etc.

These are extended by introducing, for every pair of non-negative integers a, b, with b different from 0, the fraction a/b, subject to the following conditions (which can be shown to be consistent):

- a/1 = a;
- a/b = c/d if and only if ad = bc;
- a/b R c/d if and only if ad R bc;
- a/b + c/d = (ad + bc)/bd;
- (a/b)(c/d) = ac/bd.

Then the next step is to introduce, for every non-negative rational number r, a corresponding negative rational number −r, subject to the conditions:

- −r = −s if and only if r = s;
- −r = s if and only if r = 0 and s = 0;
- −rR−s if and only if sRr;
- −rRs;
- sR−r if and only if r = 0 and s = 0;
- −r + s = s + −r = either t, where r + t = s, or −t where s + t =r;
- −r + −s = −(r + s);
- (−r)s = s(−r) = −(rs);
- (−r)(−s) = rs.

If we make the minimum extension of the system of rational numbers which will render the order continuous, the system of real numbers results. Addition and multiplication of real numbers are uniquely determined by the meanings already given to addition and multiplication of rational numbers and the requirement that addition of, or multiplication by, a fixed real number (on right or left) shall be a continuous function (see *continuity*). Subtraction and division may be introduced as inverses of addition and multiplication respectively.

Finally, the *complex numbers* are introduced as numbers a+bi, where a and b are real numbers. There is no ordering relation, but addition and multiplication are determined as follows:

(a+bi) + (c+di) = (a + c) + (b + d)i.In particular i (i.e., 0+1i) multiplied by itself is −1. A number of the form a+0i may be identified with the real number a; other complex numbers are called imaginary numbers, and those of the form 0+bi are called

(a+bi)(c+di) = (ac − bd) + (ad + bc)i.

(It is, of course, not possible to define i as "the square root of −l."
The foregoing statement corresponds to taking i as a new, undefined, symbol. But there is an alternative method, of *logical construction,* in which the complex numbers are defined as ordered pairs (a, b) of real numbers, and i is then defined as (0, 1).)

In a mathematical development of the real number system or the complex number system, an appropriate set of postulates may be the starting point. Or the non-negative integers may first be introduced (by postulates or otherwise -- see *arithmetic, foundations of*) and from these the above outlined extensions may be provided for by successive logical constructions, in any one of several alternative ways.

The important matter is not the definition of number (or of particular numbers), which may be made in various ways more or less indifferently, but the internal structure of the number *system*.

For the notions of *cardinal number, relation-number*, and *ordinal number*, see the articles of these titles. -- Alonzo Church

- R. Dedekind,
*Essays on the Theory of Numbers*, translated by W. W. Beman, Chicago, 1901.- E. V. Huntington,
*A set of postulates for real algebra*, Transactions of the American Mathematical Society, vol. 6 (1905), pp. 17-41.- E. V. Huntington,
*A set of postulates for ordinary complex algebra*, ibid., pp. 209-229.- E. Landau,
*Grundlagen der Analysis*, Leipzig, 1930.

*Fundamentos de la Anterosofia*, 1925;*Anterosofia Racional*, 1926;*De Nuevo Hablo Jesus*, 1928;*Filosofia Integral*, 1932;*Del Conocimiento y Progreso de Si Mismo*, 1934;*Tratado de Metalogica, o Fundamentos de Una Nueva Metodologia*, 1936;*Suma Contra Una Nueva Edad Media*, 1938;*Metafisica y Ciencia*, 1941;*La Honda Inquietud*, 1915;*Conocimiento y Creencia*, 1916.

Three fundamental questions and a tenacious effort to answer them run throughout the entire thought of Nuñez Regüeiro, namely the three questions of Kant: What can I know? What must I do? What can I expect? Science as auch does not write finis to anything. We experience in science the same realm of contradictions and inconsistencies which we experience elsewhere. Fundamentally, this chaos is of the nature of dysteleology. At the root of the conflict lies a crisis of values. The problem of doing is above all a problem of valuing. From a point of view of values, life ennobles itself, man lifts himself above the trammels of matter, and the world becomes meaning-*full*. Is there a possibility for the realization of this ideal? Has this plan ever been tried out? History offers us a living example: The Fact of Jesus. He is the only possible expectation. In him and through him we come to fruition and fulfilment. Nuñez Regüeiro's philosophy is fundamentally religious. -- J.A.F.