History of the function concept

2124:"There is a temptation to regard a relation as definable in extension as a class of couples. This is the formal advantage that it avoids the necessity for the primitive proposition asserting that every couple has a relation holding between no other pairs of terms. But it is necessary to give sense to the couple, to distinguish the referent from the relatum : thus a couple becomes essentially distinct from a class of two terms, and must itself be introduced as a primitive idea. . . . It seems therefore more correct to take an intensional view of relations, and to identify them rather with class-concepts than with classes." 2795:"The concept of a function which we are considering now differs essentially from the concepts of a sentential and of a designatory function .... Strictly speaking ... do not belong to the domain of logic or mathematics; they denote certain categories of expressions which serve to compose logical and mathematical statements, but they do not denote things treated of in those statements... . The term "function" in its new sense, on the other hand, is an expression of a purely logical character; it designates a certain type of things dealt with in logic and mathematics." 6516: 2343:"provide for ... the whole sweep of abstract set theory. The crux of the matter is that Schönfinkel lets functions stand as arguments. For Schönfinkel, substantially as for Frege, classes are special sorts of functions. They are propositional functions, functions whose values are truth values. All functions, propositional and otherwise, are for Schönfinkel one-place functions". Remarkably, Schönfinkel reduces all mathematics to an extremely compact 2775:, which contains variables and, on replacement of these variables by constants becomes a sentence, is called a SENTENTIAL FUNCTION. But mathematicians, by the way, are not very fond of this expression, because they use the term "function" with a different meaning. ... sentential functions and sentences composed entirely of mathematical symbols (and not words of everyday language), such as: 6102: 492:(1821). From what he says there, it is clear that he normally regards a function as being defined by an analytic expression (if it is explicit) or by an equation or a system of equations (if it is implicit); where he differs from his predecessors is that he is prepared to consider the possibility that a function may be defined only for a restricted range of the independent variable. 1068:) never stands for anything but a logical class. It may be a compound class aggregated of many simple classes; it may be a class indicated by certain inverse logical operations, it may be composed of two groups of classes equal to one another, or what is the same thing, their difference declared equal to zero, that is, a logical equation. But however composed or derived, 3126:(1922) constructed a Lebesgue integrable function whose Fourier series diverges pointwise almost everywhere. Nevertheless, a very wide class of functions can be expanded in Fourier series, especially if one allows weaker forms of convergence, such as convergence in the sense of distributions. Thus, Fourier's claim was a reasonable one in the context of his time. 1375:. The notion of the variable is one of the most difficult with which logic has to deal. For the present, I openly wish to make it plain that there are variables in all mathematical propositions, even where at first sight they might seem to be absent. . . . We shall find always, in all mathematical propositions, that the words 764:(1847) stated it: "the examination of that part of reasoning which depends upon the manner in which inferences are formed, and the investigation of general maxims and rules for constructing arguments". At this time the notion of (logical) "function" is not explicit, but at least in the work of De Morgan and 1188:

The one-argument function Frege generalizes into the form Φ(A) where A is the argument and Φ( ) represents the function, whereas the two-argument function he symbolizes as Ψ(A, B) with A and B the arguments and Ψ( , ) the function and cautions that "in general Ψ(A, B) differs from Ψ(B, A)". Using his

1144:

Frege begins his discussion of "function" with an example: Begin with the expression "Hydrogen is lighter than carbon dioxide". Now remove the sign for hydrogen (i.e., the word "hydrogen") and replace it with the sign for oxygen (i.e., the word "oxygen"); this makes a second statement. Do this again

819:

Boole asserts that "logic . . . is in a more especial sense the science of reasoning by signs", and he briefly discusses the notions of "belonging to" and "class": "An individual may possess a great variety of attributes and thus belonging to a great variety of different classes". Like De Morgan he

799:

While the word "function" does not appear, the notion of "abstraction" is there, "variables" are there, the notion of inclusion in his symbolism "all of the Δ is in the О" (p. 9) is there, and lastly a new symbolism for logical analysis of the notion of "relation" (he uses the word with respect

446:

The modern understanding of function and its definition, which seems correct to us, could arise only after Fourier's discovery. His discovery showed clearly that most of the misunderstandings that arose in the debate about the vibrating string were the result of confusing two seemingly identical but

3121:

Contemporary mathematicians, with much broader and more precise conceptions of functions, integration, and different notions of convergence than was possible in Fourier's time (including examples of functions that were regarded as pathological and referred to as "monsters" until as late as the turn

2522:

Suppes observes that von Neumann's axiomatization was modified by Bernays "in order to remain nearer to the original Zermelo system . . . He introduced two membership relations: one between sets, and one between sets and classes". Then Gödel further modified the theory: "his primitive notions are

851:

of it. The Differential calculus enables us in every case to pass from the function to the limit. This it does by a certain Operation. But in the very Idea of an Operation is . . . the idea of an inverse operation. To effect that inverse operation in the present instance is the business of the Int

3767:"In Mathematics, my chief obligations, as is indeed evident, are to Georg Cantor and Professor Peano. If I had become acquainted sooner with the work of Professor Frege, I should have owed a great deal to him, but as it is I arrived independently at many results which he had already established", 1546:

To continue the example: Suppose (from outside the mathematics/logic) one determines that the propositions "Bob is hurt" has a truth value of "falsity", "This bird is hurt" has a truth value of "truth", "Emily the rabbit is hurt" has an indeterminate truth value because "Emily the rabbit" doesn't

640:

However, Gardiner says "...it seems to me that Lakatos goes too far, for example, when he asserts that 'there is ample evidence that had no idea of concept'." Moreover, as noted above, Dirichlet's paper does appear to include a definition along the lines of what is usually ascribed to him, even

447:

actually vastly different concepts, namely that of function and that of its analytic representation. Indeed, prior to Fourier's discovery no distinction was drawn between the concepts of "function" and of "analytic representation," and it was this discovery that brought about their disconnection.

1386:

As expressed by Russell "the process of transforming constants in a proposition into variables leads to what is called generalization, and gives us, as it were, the formal essence of a proposition ... So long as any term in our proposition can be turned into a variable, our proposition can be

2095:

In this quote the reader may observe a shift in terminology: nowhere is mentioned the notion of "propositional function", but rather one sees the words "formula", "predicate calculus", "predicate", and "logical calculus." This shift in terminology is discussed more in the section that covers

2407:). The two "domains of objects" are called "arguments" (I-objects) and "functions" (II-objects); where they overlap are the "argument functions" (he calls them I-II objects). He introduces two "universal two-variable operations" – (i) the operation : ". . . read 'the value of the function 661:

Because Lobachevsky and Dirichlet have been credited as among the first to introduce the notion of an arbitrary correspondence, this notion is sometimes referred to as the Dirichlet or Lobachevsky-Dirichlet definition of a function. A general version of this definition was later used by

219:." Calculus was developed using the notion of variables, with their associated geometric meaning, which persisted well into the eighteenth century. However, the terminology of "function" came to be used in interactions between Leibniz and Bernoulli towards the end of the 17th century. 1055:

to describe "class relations", the notions "'quantifying' our predicate", "propositions in respect of their extension", "the relation of inclusion and exclusion of two classes to one another", and "propositional function" (all on p. 10), the bar over a variable to indicate

405:

The relations among these quantities are not thought of as being given by formulas, but on the other hand they are surely not thought of as being the sort of general set-theoretic, anything-goes subsets of product spaces that modern mathematicians mean when they use the word

2379:" e prefer, however, to axiomatize not "set" but "function". The latter notion certainly includes the former. (More precisely, the two notions are completely equivalent, since a function can be regarded as a set of pairs, and a set as a function that can take two values.)". 2331:"As is well known, by function we mean in the simplest case a correspondence between the elements of some domain of quantities, the argument domain, and those of a domain of function values ... such that to each argument value there corresponds at most one function value". 3139:

is a sequence of values or ordinates, each of which is arbitrary...It is by no means assumed that these ordinates are subject to any general law; they may follow one another in a completely arbitrary manner, and each of them is defined as if it were a unique quantity."

2685:

defines the words as follows: "In word languages, a proposition is expressed by a sentence. Then a 'predicate' is expressed by an incomplete sentence or sentence skeleton containing an open place. For example, "___ is a man" expresses a predicate ... The predicate is a

779:, and not upon the particular matters spoken of"; he wastes no time (preface page i) abstracting: "In the form of the proposition, the copula is made as abstract as the terms". He immediately (p. 1) casts what he calls "the proposition" (present-day propositional 2223:

1910–1913 with a further refinement called "a matrix". The first axiom is *12.1; the second is *12.11. To quote Wiener the second axiom *12.11 "is involved only in the theory of relations". Both axioms, however, were met with skepticism and resistance; see more at

2090:

of first order in which the sole predicate constants are ε and possibly, =. ... Today an axiomatization of set theory is usually embedded in a logical calculus, and it is Weyl's and Skolem's approach to the formulation of the axiom of separation that is generally

1145:(using either statement) and substitute the sign for nitrogen (i.e., the word "nitrogen") and note that "This changes the meaning in such a way that "oxygen" or "nitrogen" enters into the relations in which "hydrogen" stood before". There are three statements: 2519:, is to be regarded as a variable in this procedure". To avoid the "antinomies of naive set theory, in Russell's first of all . . . we must forgo treating certain functions as arguments". He adopts a notion from Zermelo to restrict these "certain functions". 2228:. By 1914 Norbert Wiener, using Whitehead and Russell's symbolism, eliminated axiom *12.11 (the "two-variable" (relational) version of the axiom of reducibility) by expressing a relation as an ordered pair using the null set. At approximately the same time, 631:

There is no such definition in Dirichlet's works at all. But there is ample evidence that he had no idea of this concept. In his paper for instance, when he discusses piecewise continuous functions, he says that at points of discontinuity the function

685:, which appeared in 1888 but had already been drafted in 1878. Dieudonné observes that instead of confining himself, as in previous conceptions, to real (or complex) functions, Dedekind defines a function as a single-valued mapping between any two sets: 1136:

respectively, which he believes "will stand the test of time. It is easy to see how regarding a content as a function of an argument leads to the formation of concepts. Furthermore, the demonstration of the connection between the meanings of the words

2878:"...although we do not find in the idea of functional dependence distinguished in explicit form as a comparatively independent object of study, nevertheless one cannot help noticing the large stock of functional correspondences they studied." ( 1175:

of the function "he sign , regarded as replaceable by others that denotes the object standing in these relations". He notes that we could have derived the function as "Hydrogen is lighter than . . .." as well, with an argument position on the

3597:. For most of his logical symbolism and notions of propositions Peano credits "many writers, especially Boole". In footnote 1 he credits Boole 1847, 1848, 1854, Schröder 1877, Peirce 1880, Jevons 1883, MacColl 1877, 1878, 1878a, 1880; cf 869:, from a foundation in the logic of propositions and propositional functions". But by the late 19th century the logicians' research into the foundations of mathematics was undergoing a major split. The direction of the first group, the 3964:"The nonprimitive and arbitrary character of this axiom drew forth severe criticism, and much of subsequent refinement of the logistic program lies in attempts to devise some method of avoiding the disliked axiom of reducibility" 2984:

Eves dates Leibniz's first use to the year 1694 and also similarly relates the usage to "as a term to denote any quantity connected with a curve, such as the coordinates of a point on the curve, the slope of the curve, and so on"

884:'s "set theory" (1870–1890) but were driven forward partly as a result of Russell's discovery of a paradox that could be derived from Frege's conception of "function", but also as a reaction against Russell's proposed solution. 3676:

Frege cautions that the function will have "argument places" where the argument should be placed as distinct from other places where the same sign might appear. But he does not go deeper into how to signify these positions and

1938:

allowed a function to be an argument of itself: "On the other hand, it may also be that the argument is determinate and the function indeterminate . . .." From this unconstrained situation Russell was able to form a paradox:

648:

It is a matter of some dispute how much credit Dirichlet deserves for the modern definition of a function, in part because he restricted his definition to continuous functions....I believe Dirichlet defined the notion of

531:. The value of the function can be given either by an analytic expression, or by a condition that provides a means of examining all numbers and choosing one of them; or finally the dependence may exist but remain unknown. 1542:

is all objects satisfying some propositional function" (p. 23). Note the word "all" – this is how the contemporary notions of "For all ∀" and "there exists at least one instance ∃" enter the treatment (p. 15).

1417:

does not belong to the function but the two taken together make the whole". Russell agreed with Frege's notion of "function" in one sense: "He regards functions – and in this I agree with him – as more fundamental than

163:

for the equation to have a solution. He then determined the maximum value of this expression. It is arguable that the isolation of this expression is an early approach to the notion of a "function". A value less than

1791:

Hilbert then illustrates the three ways how the ε-function is to be used, firstly as the "for all" and "there exists" notions, secondly to represent the "object of which holds", and lastly how to cast it into the

1510:; this proposition is called a "value" of the propositional function. In our example there are four values of the propositional function, e.g., "Bob is hurt", "This bird is hurt", "Emily the rabbit is hurt" and " 1955:

Frege responded promptly that "Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic".

1180:; the exact observation is made by Peano (see more below). Finally, Frege allows for the case of two (or more) arguments. For example, remove "carbon dioxide" to yield the invariant part (the function) as: 2783:= 5 are usually referred to by mathematicians as FORMULAE. In place of "sentential function" we shall sometimes simply say "sentence" – but only in cases where there is no danger of any misunderstanding". 65: 75:-coordinate of the point. Functions were not explicitly considered in antiquity, but some precursors of the concept can perhaps be seen in the work of medieval philosophers and mathematicians such as 768:

it is implied: we see abstraction of the argument forms, the introduction of variables, the introduction of a symbolic algebra with respect to these variables, and some of the notions of set theory.

247:

started calling expressions made of a single variable "functions." In 1698, he agreed with Leibniz that any quantity formed "in an algebraic and transcendental manner" may be called a function of

1854:

that can be calculated by an algorithm. The outcomes of these efforts were vivid demonstrations that, in Turing's words, "there can be no general process for determining whether a given formula

1360:, "...a point in which Frege's work is very important, and requires careful examination". In response to his 1902 exchange of letters with Frege about the contradiction he discovered in Frege's 5294:

With commentary by van Heijenoort. Wherein Zermelo attempts to solve Russell's paradox by structuring his axioms to restrict the universal domain B (from which objects and sets are pulled by

401:

Medvedev considers that "In essence this is the definition that became known as Dirichlet's definition." Edwards also credits Euler with a general concept of a function and says further that

2192:

But he goes on to say that he would not introduce the ordered couples further into his "symbolic treatment"; he proposes his "matrix" and his unpopular axiom of reducibility in their place.

1193:"We can read |--- Φ(A) as "A has the property Φ. |--- Ψ(A, B) can be translated by "B stands in the relation Ψ to A" or "B is a result of an application of the procedure Ψ to the object A". 5404:

With commentary by van Heijenoort. Wherein von Neumann creates "classes" as distinct from "sets" (the "classes" are Zermelo's "definite properties"), and now there is a universal set, etc.

877:– "to fulfil two objects, first, to show that all mathematics follows from symbolic logic, and secondly to discover, as far as possible, what are the principles of symbolic logic itself." 3122:

of the 20th century), would not agree with Fourier that a completely arbitrary function can be expanded in Fourier series, even if its Fourier coefficients are well-defined. For example,

4357:

von Neumann's critique of the history observes the split between the logicists (e.g., Russell et al.) and the set-theorists (e.g., Zermelo et al.) and the formalists (e.g., Hilbert), cf

1959:

From this point forward development of the foundations of mathematics became an exercise in how to dodge "Russell's paradox", framed as it was in "the bare notions of set and element".

1943:"You state ... that a function, too, can act as the indeterminate element. This I formerly believed, but now this view seems doubtful to me because of the following contradiction. Let 2371:(1922) had amended Zermelo's set theory of 1908. But von Neumann was not convinced that this axiomatization could not lead to the antinomies. So he proposed his own theory, his 1925 653:

function to make it clear that no rule or law is required even in the case of continuous functions, not just in general. This would have deserved special emphasis because of Euler's

2690:. Predicates are often called 'properties' ... The predicate calculus will treat of the logic of predicates in this general sense of 'predicate', i.e., as propositional function". 1458:

Russell, like Frege, considered the propositional function fundamental: "Propositional functions are the fundamental kind from which the more usual kinds of function, such as "sin

733:. This broad definition of a function encompasses more relations than are ordinarily considered functions in contemporary mathematics. For example, Hardy's definition includes 1787:

13. A(a) --> A(ε(A)) Here ε(A) stands for an object of which the proposition A(a) certainly holds if it holds of any object at all; let us call ε the logical ε-function".

211:

The development of analytical geometry around 1640 allowed mathematicians to go between geometric problems about curves and algebraic relations between "variable coordinates

847:"That quantity whose variation is uniform . . . is called the independent variable. That quantity whose variation is referred to the variation of the former is said to be a 1822:", that is, an explicit, step-by-step procedure that would succeed in computing a function. Various models for algorithms appeared, in rapid succession, including Church's 1470:" are derived. These derivative functions . . . are called "descriptive functions". The functions of propositions . . . are a particular case of propositional functions". 369: 2275:

reduce nor otherwise change the propositional-function form *12.1; indeed he declared this "essential to the treatment of identity, descriptions, classes and relations".

301:

and derivatives; both are measurements of the output or the change in the output as it depends on the input or the change in the input. Such functions are the basis of

191:. Nevertheless, Medvedev suggests that the implicit concept of a function is one with an ancient lineage. Ponte also sees more explicit approaches to the concept in the 1649:: NOT("Bob is hurt") AND "This bird is hurt". To determine the truth value of this "function of propositions with arguments" we submit it to a "truth function", e.g., 594: 1478:: Because his terminology is different from the contemporary, the reader may be confused by Russell's "propositional function". An example may help. Russell writes a 3867:. Tarski refers to a "relational function" as a "ONE-MANY or FUNCTIONAL RELATION or simply a FUNCTION". Tarski comments about this reversal of variables on page 99. 1967:

The notion of "function" appears as Zermelo's axiom III—the Axiom of Separation (Axiom der Aussonderung). This axiom constrains us to use a propositional function Φ(

288: 2211:, the purpose of which were to reduce (single-variable) propositional functions and (dual-variable) relations to a "lower" form (and ultimately into a completely 2136:

definition of a relation, stating that "mathematics is always concerned with extensions rather than intensions" and "Relations, like classes, are to be taken in

861:

Eves observes "that logicians have endeavored to push down further the starting level of the definitional development of mathematics and to derive the theory of

4403:

Bernays asserts (in the context of rebutting Logicism's construction of the numbers from logical axioms) that "the Number concept turns out to be an elementary

1634:: "This bird is hurt". (We are restricted to the logical linkages NOT, AND, OR and IMPLIES, and we can only assign "significant" propositions to the variables 5572:

Monna, A. F. (1972). "The concept of function in the 19th and 20th centuries, in particular with regard to the discussions between Baire, Borel and Lebesgue".

616:

by one and the same law throughout the entire interval, and it is not necessary that it be regarded as a dependence expressed using mathematical operations.

6241: 375:

A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.

2791:

calls the relational form of function a "FUNCTIONAL RELATION or simply a FUNCTION". After a discussion of this "functional relation" he asserts that:

2399:

that are I-objects (first axiom), and two types of "operations" that assume ordering as a structural property obtained of the resulting objects and (

199:

Historically, some mathematicians can be regarded as having foreseen and come close to a modern formulation of the concept of function. Among them is

4939: 4869: 3156:, p. 263. Translation by Abe Shenitzer of an article by Luzin that appeared (in the 1930s) in the first edition of The Great Soviet Encyclopedia 6159: 1076:) with us will never be anything else than a general expression for such logical classes of things as may fairly find a place in ordinary Logic". 397:

of the second. This name has an extremely broad character; it encompasses all the ways in which one quantity can be determined in terms of others.

3333: 4965: 2621:, and the function is said to be determined by the given functional relation. Two equivalent functional relations determine the same function." 2524: 6391: 6226: 4899: 4661: 1206:

Peano defined the notion of "function" in a manner somewhat similar to Frege, but without the precision. First Peano defines the sign "K means

2865:"The emergence of a notion of function as an individualized mathematical entity can be traced to the beginnings of infinitesimal calculus". ( 1371:: "6. Mathematical propositions are not only characterized by the fact that they assert implications, but also by the fact that they contain 430:

nor defined by an analytical expression. Related questions on the nature and representation of functions, arising from the solution of the

2032:– "...this disposes of the Russell antinomy so far as we are concerned". But Zermelo's "definite criterion" is imprecise, and is fixed by 4500:, p. 12 footnote. He also references "a paper by R. M. Robinson provides a simplified system close to von Neumann's original one". 620:

Eves asserts that "the student of mathematics usually meets the Dirichlet definition of function in his introductory course in calculus.

2693:

In 1954, Bourbaki, on p. 76 in Chapitre II of Theorie des Ensembles (theory of sets), gave a definition of a function as a triple

1270:

denotes a new object". Peano adds two conditions on these new objects: First, that the three equality-conditions hold for the objects φ

1060:(page 43), etc. Indeed he equated unequivocally the notion of "logical function" with "class" : "... on the view adopted in this book, 657:

of a continuous function as one given by single expression-or law. But I also doubt there is sufficient evidence to settle the dispute.

456:

During the 19th century, mathematicians started to formalize all the different branches of mathematics. One of the first to do so was

6432: 2291:

states that "It should be observed that all mathematical functions result form one-many relations . . . Functions in this sense are

393:

When certain quantities depend on others in such a way that they undergo a change when the latter change, then the first are called

3710:

and his subsequent works, but he does not till that ground to any depth comparable to what Frege does in his self-allotted field",

4635: 2108:" is not clear. As noted above, Frege (1879) proposed an intuitive ordering in his definition of a two-argument function Ψ(A, B). 6458: 2967: 1772:

set himself the goal of "formalizing" classical mathematics "as a formal axiomatic theory, and this theory shall be proved to be

1302:". Given all these conditions are met, φ is a "function presign". Likewise he identifies a "function postsign". For example if 34: 6336: 1387:

generalized; and so long as this is possible, it is the business of mathematics to do it"; these generalizations Russell named

6126: 677:, who was one of the founding members of the Bourbaki group, credits a precise and general modern definition of a function to 6545: 5446: 4859: 4837: 3937: 3010: 2941: 1807: 6152: 384: 183:

According to Dieudonné and Ponte, the concept of a function emerged in the 17th century as a result of the development of

1350:

While the influence of Cantor and Peano was paramount, in Appendix A "The Logical and Arithmetical Doctrines of Frege" of

6453: 6214: 5758: 315: 6386: 6351: 5887: 2675:) (in Halmos), they will see no mention of "proposition" or even "first order predicate calculus". In their place are " 6036: 5609: 5472: 5114: 5076: 5047: 5011: 4955: 4809: 4783: 4745: 4645: 4484: 4416: 3343: 3315: 1863: 6494: 3776: 2347:

consisting of only three functions: Constancy, fusion (i.e., composition), and mutual exclusivity. Quine notes that

2113: 6482: 6363: 6316: 6120: 2523:

those of set, class and membership (although membership alone is sufficient)". This axiomatization is now known as

1764: 6550: 6540: 6519: 6448: 6145: 5712: 5272:

With commentary by van Heijenoort. Wherein Zermelo rails against Poincaré's (and therefore Russell's) notion of

4473:

The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory

689:

What was new and what was to be essential for the whole of mathematics was the entirely general conception of a

6309: 6297: 6041: 5540: 4669: 1419: 505: 2819:

Katz, Victor; Barton, Bill (October 2007). "Stages in the History of Algebra with Implications for Teaching".

843:

In the context of "the Differential Calculus" Boole defined (circa 1849) the notion of a function as follows:

666:(1939), and some in the education community refer to it as the "Dirichlet–Bourbaki" definition of a function. 319:, published in 1748, Euler gave essentially the same definition of a function as his teacher Bernoulli, as an 6463: 6292: 6231: 5215: 2278: 2268:"This definition . . . was historically important in reducing the theory of relations to the theory of sets. 1352: 2051:

In fact Skolem in his 1922 referred to this "definite criterion" or "property" as a "definite proposition":

6412: 3923: 1426:" but Russell rejected Frege's "theory of subject and assertion", in particular "he thinks that, if a term 106:, eventually led to the much more general modern concept of a function as a single-valued mapping from one 5174:

With commentary by van Heijenoort. Wherein Russell announces his discovery of a "paradox" in Frege's work.

2721:) Bourbaki states (literal translation): "Often we shall use, in the remainder of this Treatise, the word 1902:'s attempt to define the infinite in set-theoretic treatment (1870–1890) and a subsequent discovery of an 1585:

is hurt", only "This bird" is included, given the four values "Bob", "This bird", "Emily the rabbit" and "

6116: 2141: 473: 207:

In his theory, some general ideas about independent and dependent variable quantities seem to be present.

5616:

Ruthing, D. (1984). "Some definitions of the concept of function from Bernoulli, Joh. to Bourbaki, N.".

2634:

Both axiomatic and naive forms of Zermelo's set theory as modified by Fraenkel (1922) and Skolem (1922)

2539:, in addition to giving the well-known ordered pair definition of a function as a certain subset of the 6067: 4829: 3929:

The undecidable: basic papers on undecidable propositions, unsolvable problems and computable functions

3876:

Whitehead and Russell 1910–1913:31. This paper is important enough that van Heijenoort reprinted it as

820:

uses the notion of "variable" drawn from analysis; he gives an example of "represent the class oxen by

508:

are traditionally credited with independently giving the modern "formal" definition of a function as a

1878:

Set theory began with the work of the logicians with the notion of "class" (modern "set") for example

1745:

function. All the ordinary functions of mathematics are of this kind. Thus in our notation "sin

771:

De Morgan's 1847 "FORMAL LOGIC OR, The Calculus of Inference, Necessary and Probable" observes that "

251:. By 1718, he came to regard as a function "any expression made up of a variable and some constants." 6555: 6324: 6046: 5971: 5753: 1851: 176:

corresponds to two solutions. Sharaf al-Din's analysis of this equation was a notable development in

4585:

An Investigation into the Laws of Thought on which are founded the Laws of Thought and Probabilities

4467:

cf in particular p. 35 where Gödel declares his primitive notions to be class, set, and "the dyadic

4374:

In addition to the 1925 appearance in van Heijenoort, Suppes 1970:12 cites two more: 1928a and 1929.

1838:'s (1936–7) notion of replacing human "computers" with utterly-mechanical "computing machines" (see 180:, but his work was not pursued any further at that time, neither in the Muslim world nor in Europe. 6253: 5944: 2972: 2120:

considered the definition of a relation (such as Ψ(A, B)) as a "class of couples" but rejected it:

320: 5529:"Between rigor and applications: Developments in the concept of function in mathematical analysis" 2962: 1159:

Now observe in all three a "stable component, representing the totality of relations"; call this

330: 6287: 6275: 6246: 6172: 6028: 3287:

of the function . . . it stresses the basic idea of a relationship between two sets of numbers"

2601:. We give the name of function to the operation which in this way associates with every element 2099: 6407: 6341: 6202: 6168: 5428: 5303: 4969: 2287:

notion of "function" as a many-one correspondence derives from is unclear. Russell in his 1920

2271:

Observe that while Wiener "reduced" the relational *12.11 form of the axiom of reducibility he

2216: 1883: 1474: 1423: 1388: 1368: 1114: 509: 379:

Euler also allowed multi-valued functions whose values are determined by an implicit equation.

252: 188: 24: 6489: 5528: 5360:

With commentary by van Heijenoort. Wherein Skolem defines Zermelo's vague "definite property".

3000: 2375:. It explicitly contains a "contemporary", set-theoretic version of the notion of "function": 787:) into a form such as "X is Y", where the symbols X, "is", and Y represent, respectively, the 562: 476:). According to Smithies, Cauchy thought of functions as being defined by equations involving 6501: 6356: 6331: 6263: 6209: 6082: 5680: 5645:

Youschkevitch, A. P. (1976). "The concept of function up to the middle of the 19th century".

4769: 2929: 2225: 2208: 2133: 1923: 1831: 1814:, mathematicians set about to define what was meant by an "effectively calculable function" ( 1109: 426:. Fourier had a general conception of a function, which included functions that were neither 119: 87: 3472:

Elementary Treatise on Logic not mathematical including philosophy of mathematical reasoning

262: 6381: 6280: 6001: 5847: 4699: 1915: 1911: 1867: 1811: 1210:, or aggregate of objects", the objects of which satisfy three simple equality-conditions, 899: 893: 738: 734: 5365: 4798: 3190:Über die Darstellung ganz willkürlicher Funktionen durch Sinus- und Cosinusreihen," 1837 ( 2679:

of the object language", "atomic formulae", "primitive formulae", and "atomic sentences".

2324: 1555:

itself is ambiguous. While the two propositions "Bob is hurt" and "This bird is hurt" are

8: 6422: 6270: 6221: 6192: 5956: 5939: 5919: 5882: 5831: 5826: 5768: 5705: 5551:

Malik, M. A. (1980). "Historical and pedagogical aspects of the definition of function".

5100: 4881: 4847: 2958: 2083: 2075: 1907: 1843: 641:

though (like Lobachevsky) he states it only for continuous functions of a real variable.

501: 427: 298: 177: 83: 68: 4989: 2717:, meaning a set of pairs where no two pairs have the same first member. On p. 77 ( 1413:

is taken away, i.e., in the above instance 2( ) + ( ). The argument

82:

Mathematicians of the 18th century typically regarded a function as being defined by an

6427: 6187: 5892: 5821: 5778: 5662: 5633: 5589: 5515: 5487:

Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens

4944: 4927: 4893: 4655: 4593: 2836: 2148:: "We may regard a relation ... as a class of couples ... the relation determined by φ( 2087: 1847: 1664:( NOT("Bob is hurt") AND "This bird is hurt" ), which yields a truth value of "truth". 866: 761: 438:

and Euler, and they had a significant impact in generalizing the notion of a function.

5130:

Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought

1873: 1611:. For example, suppose one were to form the "function of propositions with arguments" 6417: 6106: 6077: 6072: 6062: 5996: 5924: 5809: 5666: 5637: 5605: 5593: 5536: 5468: 5442: 5110: 5072: 5043: 5007: 4951: 4855: 4833: 4805: 4779: 4741: 4641: 4480: 4412: 3933: 3339: 3311: 3006: 2937: 2840: 2561:

be two sets, which may or may not be distinct. A relation between a variable element

2540: 2352: 1765:

The formalist's "function": David Hilbert's axiomatization of mathematics (1904–1927)

862: 742: 228: 184: 107: 6137: 4619: 1107:. Russell in turn influenced much of 20th-century mathematics and logic through his 674: 6304: 6011: 5737: 5732: 5654: 5625: 5581: 5560: 5507: 5432: 5391: 5307: 5237: 5161: 4999: 4985: 4919: 4685: 4607: 4522: 4471:

between class and class, class and set, set and class, or set and set". Gödel 1940

2828: 2536: 2364: 2160:) is true". In a footnote he clarified his notion and arrived at this definition: 2045: 2037: 1713:

Russell symbolizes the descriptive function as "the object standing in relation to

663: 461: 244: 5495: 4690: 4673: 3551:, pp. 31–34. Boole discusses this "special law" with its two algebraic roots 2112:

in his 1914 (see below) observes that his own treatment essentially "revert(s) to

1441: 5857: 5799: 5436: 5229: 5104: 5066: 5037: 4823: 4773: 4759: 4735: 4695: 4597: 4583: 4202: 3927: 3626:

Frege's exact words are "expressed in our formula language" and "expression", cf

3305: 2229: 1919: 1914:), by the discovery of more antinomies in the early 20th century (e.g., the 1897 1823: 1793: 1534:. If a proposition's truth value is "truth" then the variable's value is said to 1393: 1092: 488:

Cauchy makes some general remarks about functions in Chapter I, Section 1 of his

5090: 2755:

The reason for the disappearance of the words "propositional function" e.g., in

1922:), and by resistance to Russell's complex treatment of logic and dislike of his 6111: 5804: 5783: 5698: 5533:

The Cambridge History of Science: The modern physical and mathematical sciences

5347: 5329: 5298:) so that it itself cannot be a set, i.e., his axioms disallow a universal set. 5143: 5033: 5021: 4755: 4719: 4707: 4458:

See also van Heijenoort's introduction to von Neumann's paper on pages 393–394.

2368: 2109: 1839: 1827: 1693:

The reader should be warned here that the order of the variables are reversed!

1096: 481: 423: 419: 256: 2832: 2055:"... a finite expression constructed from elementary propositions of the form 1383:

occur; and these words are the marks of a variable and a formal implication".

832:

by the sign + . . . we might represent the aggregate class oxen and horses by

6534: 5961: 5902: 5564: 5482: 5409: 5379: 5321: 5281: 5273: 5259: 5251: 5197: 5179: 5125: 5062: 4907: 3500: 2800: 2788: 2348: 2336: 2025: 1947:

be the predicate: to be a predicate that cannot be predicated of itself. Can

1815: 1769: 1593:

and their respective truth-values: falsity, truth, indeterminate, ambiguous.

1088: 772: 439: 435: 431: 200: 4256:, p. 32. This same point appears in van Heijenoort's commentary before 3251:

there is automatically assigned, by some rule or correspondence, a value to

2629: 2199:

led Russell to propose his "doctrine of types" in an appendix B of his 1903

1027:

Venn was using the words "logical function" and the contemporary symbolism (

5951: 5773: 5684: 4579: 3301: 2252:(1921) offered a definition that has been widely used ever since, namely {{ 2105: 2033: 1899: 1783:

he frames the notion of function in terms of the existence of an "object":

881: 765: 624: 103: 4970:"The history of the concept of function and some educational implications" 3239:

is a symbol that represents any one of a set of numbers; if two variables

2747:(p. 86) as a relation where no two pairs have the same first member. 1842:). It was shown that all of these models could compute the same class of 1682:

that satisfies the (2-variable) propositional function (i.e., "relation")

5986: 5981: 5934: 5553:

International Journal of Mathematical Education in Science and Technology

4819: 4793: 4731: 2212: 1989:"AXIOM III. (Axiom of separation). Whenever the propositional function Φ( 1835: 1773: 477: 192: 91: 20: 2082:"A property is definite in Skolem's sense if it is expressed . . . by a 1962: 1341: 1103:

until after he had published his 1889. Both writers strongly influenced

705:, pp. 26–28 defined a function as a relation between two variables 460:; his somewhat imprecise results were later made completely rigorous by 434:

for a vibrating string, had already been the subject of dispute between

5929: 5897: 5862: 5658: 5629: 5585: 5519: 4931: 3235:

Eves asserts that Dirichlet "arrived at the following formulation: " a

3123: 3002:

Elements of Mathematics Functions of a Real Variable: Elementary Theory

2249: 2196: 2184:. We shall call it a "couple with sense," ... it may also be called an 1926:(1908, 1910–1913) that he proposed as a means to evade the antinomies. 1258:). He then introduces φ, "a sign or an aggregate of signs such that if 1189:

unique symbolism he translates for the reader the following symbolism:

465: 294: 236: 99: 2279:

Schönfinkel's notion of "function" as a many-one "correspondence" 1924

1934:

In 1902 Russell sent a letter to Frege pointing out that Frege's 1879

1051:), cf page xxi) plus the circle-diagrams historically associated with 5991: 5852: 5763: 5086: 4631: 2203:. In a few years he would refine this notion and propose in his 1908 1819: 1430:

occurs in a proposition, the proposition can always be analysed into

1052: 757: 721:." He neither required the function to be defined for all values of 5511: 4923: 3779:'s three volumes of "non-Peanesque methods" 1890, 1891, and 1895 cf 748: 5912: 5106:

From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931

3706:"...Peano intends to cover much more ground than Frege does in his 2100:

The Wiener–Hausdorff–Kuratowski "ordered pair" definition 1914–1921

1903: 1670:: Russell first discusses the notion of "identity", then defines a 870: 753: 678: 469: 302: 95: 28: 5068:

Introduction to Logic and to the Methodology of Deductive Sciences

4599:

Formal Logic, or The Calculus of Inference, Necessary and Probable

2934:

Who Gave You the Epsilon?: And Other Tales of Mathematical History

1538:

the propositional function. Finally, per Russell's definition, "a

231:, in a 1673 letter, to describe a quantity related to points of a 86:. In the 19th century, the demands of the rigorous development of 27:

dates from the 17th century in connection with the development of

5976: 5907: 4172: 4170: 4116:

The notion "definite" and the independence of the axiom of choice

1910:), by Russell's discovery (1902) of an antinomy in Frege's 1879 ( 1810:

of 1931. At about the same time, in an effort to solve Hilbert's

1803: 1551:

is hurt" is ambiguous as to its truth value because the argument

1494:). For our example, we will assign just 4 values to the variable 885: 324: 222: 159:

stating that the left hand side must at least equal the value of

896:; here too the notion of "propositional function" plays a role. 856: 5814: 3734: 3732: 2041: 2028:— sets originate by way of Axiom II from elements of (non-set) 1972: 880:

The second group of logicians, the set-theorists, emerged with

457: 76: 4775:

George Boole: Selected Manuscripts on Logic and its Philosophy

4167: 3275:

is called the dependent variable. The permissible values that

873:, can probably be summed up best by Bertrand Russell 519:

The general concept of a function requires that a function of

6006: 5721: 5465:

The Concept of Function: Aspects of Epistemology and Pedagogy

4048:

The original uses an Old High German symbol in place of Φ cf

3335:

Understanding infinity, the mathematics of infinite processes

3283:

of the function, and the values taken on by y constitute the

2439:

must both be arguments and that itself produces an argument (

2415:. . . it itself is a type I object", and (ii) the operation ( 1490:

is hurt". (Observe the circumflex or "hat" over the variable

623:

Dirichlet's claim to this formalization has been disputed by

422:

claimed that an arbitrary function could be represented by a

240: 232: 4201:"*12 The Hierarchy of Types and the axiom of Reducibility". 3729: 3717: 2232:(1914, p. 32) gave the definition of the ordered pair ( 1559:(both have truth values), only the value "This bird" of the 1197: 924:"8. Definition. – Any algebraic expression involving symbol 760:(the 2000-year-old Aristotelian forms and otherwise), or as 5966: 3499:

Some of this criticism is intense: see the introduction by

2503:)". To clarify the function pair he notes that "Instead of 60:{\displaystyle \operatorname {d} \!y/\operatorname {d} \!x} 4503: 4182: 1802:: But the unexpected outcome of Hilbert's and his student 512:

in which every first element has a unique second element.

3793: 3791: 3789: 1874:

Development of the set-theoretic definition of "function"

1642:). Then the "function of propositions with arguments" is 811:" A Y)X To take a Y it is sufficient to take a X" , etc. 293:

The functions considered in those times are called today

5690: 4407:". This paper appears on page 243 in Paolo Mancosu 1998 4401:

The Philosophy of Mathematics and Hilbert's Proof Theory

3414: 3005:. Springer Science & Business Media. pp. 154–. 2932:. In Marlow Anderson; Victor Katz; Robin Wilson (eds.). 2116:

treatment of a relation as a class of ordered couples".

1741:, but not a propositional function ; we shall call it a 608:

for this interval. It is not at all necessary here that

472:, which favoured Euler's definition over Leibniz's (see 172:

corresponds to one solution, while a value greater than

5148:

The principles of arithmetic, presented by a new method

3159: 1356:, Russell arrives at a discussion of Frege's notion of 1120:

At the outset Frege abandons the traditional "concepts

5547:

An approachable and diverting historical presentation.

4768: 4557: 3803: 3786: 3475: 3458: 3445: 3426: 3310:. Cambridge: Cambridge University Press. p. 151. 3079: 2909: 2897: 2847: 2767:

together with further explanation of the terminology:

2327:

expressed the notion, claiming it to be "well known":

1167:"... is lighter than carbon dioxide", is the function. 6167: 5220:

The principles of mathematics and the problem of sets

4545: 4283: 4142: 3827: 3571: 3091: 1963:

Zermelo's set theory (1908) modified by Skolem (1922)

1442:

Evolution of Russell's notion of "function" 1908–1913

565: 333: 265: 37: 4533: 4114:

van Heijenoort's introduction to Abraham Fraenkel's

3946: 3890: 3815: 3352: 3247:

are so related that whenever a value is assigned to

2646:

an ordered pair as a set of two "dissymetric" sets.

2351:(1958) carried this work forward "under the head of 1850:

holds that this class of functions exhausts all the

1502:". Substitution of one of these values for variable 1448:

Mathematical logical as based on the theory of types

756:

of this time were primarily involved with analyzing

113: 5506:(4). Mathematical Association of America: 282–300. 5496:"Evolution of the Function Concept: A Brief Survey" 5109:(3rd printing ed.). Harvard University Press. 4737:

Foundations and Fundamental Concepts of Mathematics

4081:cf van Heijenoort's commentary before Zermelo 1908 3529: 3481: 3103: 3067: 3031: 3019: 2930:"Evolution of the Function Concept: A Brief Survey" 1364:Russell tacked this section on at the last moment. 5242:Mathematical logic as based on the theory of types 5026:Cauchy and the Creation of Complex Function Theory 4943: 4797: 4445:This notion is not easy to summarize; see more at 3508:Mathematical logic as based on the theory of types 3177:"On the vanishing of trigonometric series," 1834 ( 3055: 2799:See more about "truth under an interpretation" at 1446:Russell would carry his ideas forward in his 1908 1401:: That "the essence of the arithmetical function 2 916:Boole now defined a function in terms of a symbol 588: 363: 282: 59: 5286:Investigations in the foundations of set theory I 5264:A new proof of the possibility of a well-ordering 4083:Investigations in the foundations of set theory I 3854:Whitehead and Russell 1910–1913:6, 8 respectively 3387:we understand a law that assigns to each element 2956: 932:, and may be represented by the abbreviated form 890:Investigations in the foundations of set theory I 800:to this example " X)Y " (p. 75) ) is there: 389:Euler gave a more general concept of a function: 53: 41: 6532: 5427: 5301: 4712:Introduction to Analysis of the Infinite. Book I 4610:; Pogorzelski, H.; Ryan, W.; Snyder, W. (1995). 4176: 4161: 3877: 3771:, p. viii. He also highlights Boole's 1854 2885: 2750: 2358: 2008:containing as elements precisely those elements 297:. For this type of function, one can talk about 227:The term "function" was literally introduced by 4527:Elements de Mathematique, Theorie des Ensembles 2630:Notion of "function" in contemporary set theory 2132:Russell had given up on the requirement for an 1668:The notion of a "many-one" functional relation" 1498:: "Bob", "This bird", "Emily the rabbit", and " 1397:and presents a vivid example from Frege's 1891 1079: 808:X)Y To take an X it is necessary to take a Y" 5099: 4446: 4433: 4387: 4362: 4345: 4328: 4308: 4277: 4261: 4241: 4221: 4136: 4119: 4102: 4086: 4069: 4053: 4036: 4019: 4000: 3981: 3909: 3884: 3755: 3738: 3723: 3711: 3694: 3664: 3648: 3631: 3614: 3598: 3594: 3523: 3511: 2663:observes the use of function-symbolism in the 2140:". To demonstrate the notion of a relation in 6153: 5706: 5644: 5462: 5441:(1962 ed.). Cambridge University Press. 4678:Bulletin of the American Mathematical Society 1753:", and "sin" would stand for the relation sin 1577:is hurt". When one goes to form the class α: 1367:For Russell the bedeviling notion is that of 168:means no positive solution; a value equal to 71:at a point was regarded as a function of the 5370:On the building blocks of mathematical logic 4325:On the building blocks of mathematical logic 4305:On the building blocks of mathematical logic 2992: 2315:): "the singular object that has a relation 2295:functions". A reasonable possibility is the 1906:(contradiction, paradox) in this treatment ( 496: 484:, and tacitly assumed they were continuous: 313:In the first volume of his fundamental text 5389: 5363: 4880: 4429: 4383: 4358: 4321: 4301: 3932:. Courier Dover Publications. p. 145. 3515: 3338:. Courier Dover Publications. p. 275. 3304:(1976). Worrall, John; Zahar, Elie (eds.). 3203: 3178: 2998: 2921: 2340: 551:ranges continuously over the interval from 118:Already in the 12th century, mathematician 98:in terms of analysis, and the invention of 6160: 6146: 5713: 5699: 5334:A simplification of the logic of relations 5202:On the foundations of logic and arithmetic 4898:: CS1 maint: location missing publisher ( 4660:: CS1 maint: location missing publisher ( 4521: 4238:A simplification of the logic of relations 4131:But Wiener offers no date or reference cf 2818: 2642:a relation as a set of ordered pairs, and 1391:. Indeed he cites and quotes from Frege's 1155:"Nitrogen is lighter than carbon dioxide." 1149:"Hydrogen is lighter than carbon dioxide." 4946:Scenes from the History of Real Functions 4689: 4630: 4618: 4612:What are Numbers and What Should They Be? 4592: 4031:van Heijenoort's commentary to Russell's 3432: 3420: 3219: 3191: 2853: 1929: 1879: 1626:" and assign its variables the values of 948:expressions to define both algebraic and 914:An Investigation into the laws of thought 5279: 5235: 5020: 4938: 4606: 4475:appearing on pages 33ff in Volume II of 4065: 4049: 3504: 3408: 3395:a uniquely determined object called the 3374:, pp. 55–70 for further discussion. 3371: 3331: 3223: 3207: 3165: 3085: 2915: 2903: 2879: 1993:) is definite for all elements of a set 1598:functions of propositions with arguments 1152:"Oxygen is lighter than carbon dioxide." 327:involving variables and constants e.g., 5615: 5493: 5467:. Mathematical Association of America. 5407: 5310:(1967) . "Whitehead and Russell (1910) 5257: 5228:With commentary by van Heijenoort. The 5213: 5195: 5159: 5004:Introduction to Mathematical Philosophy 4998: 4984: 4754: 4668: 4289: 4233:commentary by van Heijenoort preceding 4188: 4148: 3993: 3833: 3821: 3809: 3797: 3780: 3768: 3678: 3300: 3141: 3109: 3097: 2968:MacTutor History of Mathematics Archive 2927: 2447:). Its most important property is that 2289:Introduction to Mathematical Philosophy 2195:An attempt to solve the problem of the 2117: 2096:"function" in contemporary set theory. 1777: 1450:and into his and Whitehead's 1910–1913 1104: 888:'s set-theoretic response was his 1908 874: 749:The logician's "function" prior to 1850 547:, and moreover in such a way that when 6533: 5604:, Dover Publishing Inc., New York NY, 5526: 5463:Dubinsky, Ed; Harel, Guershon (1992). 5352:Some remarks on axiomatized set theory 5345: 5327: 5061: 5032: 4868: 4846: 4792: 4674:"Euler's definition of the derivative" 4563: 4551: 4539: 4509: 4497: 4273: 4257: 4253: 4234: 4217: 4132: 4098: 3952: 3896: 3864: 3358: 2764: 2760: 2756: 2743:(p. 57) as a set of pairs, and a 2732: 2688:propositional function of one variable 2682: 2657: 2650: 2265: 1294:, we assume it is possible to deduce φ 1139:if, and, not, or, there is, some, all, 1099:(1889), but Peano had no knowledge of 523:be defined as a number given for each 6141: 5694: 5647:Archive for History of Exact Sciences 5574:Archive for History of Exact Sciences 5571: 5550: 5177: 5141: 5123: 4964: 4906: 4818: 4718: 4706: 4614:. Research Institute for Mathematics. 4578: 4012: 3977: 3922: 3751: 3690: 3660: 3644: 3627: 3610: 3590: 3548: 3535: 3153: 3073: 3061: 2891: 2866: 2812: 1895: 1891: 1776:, i.e., free from contradiction". In 1100: 702: 464:, who advocated building calculus on 5085: 4730: 4724:Foundations of Differential Calculus 4320:commentary by W. V. Quine preceding 3965: 3589:cf van Heijenoort's introduction to 3577: 3565: 3564:Although he gives others credit, cf 3487: 3288: 3049: 3037: 3025: 2986: 2617:which is in the given relation with 2597:which is in the given relation with 2525:von Neumann–Bernays–Gödel set theory 1887: 1818:1936), i.e., "effective method" or " 1322:, or if φ is the function postsign + 385:Institutiones calculi differentialis 223:The notion of "function" in analysis 5888:Analytic and synthetic propositions 5759:Formal semantics (natural language) 4764:. Paris: Firmin Didot Père et Fils. 2577:is called a functional relation in 2299:notion of "descriptive function" – 2219:would carry this treatment over to 2144:Russell now embraced the notion of 1141:and so forth, deserves attention". 857:The logicians' "function" 1850–1950 316:Introductio in analysin infinitorum 13: 5456: 5422:With commentary by van Heijenoort. 5342:With commentary by van Heijenoort. 5210:With commentary by van Heijenoort. 5192:With commentary by van Heijenoort. 5156:With commentary by van Heijenoort. 5138:With commentary by van Heijenoort. 4854:. North-Holland (published 1971). 3883:with commentary by W. V. Quine in 3476:Grattan-Guinness & Bornet 1997 3459:Grattan-Guinness & Bornet 1997 3446:Grattan-Guinness & Bornet 1997 2821:Educational Studies in Mathematics 2423:): ". . . (read 'the ordered pair 1800:Recursion theory and computability 1514:is hurt." A proposition, if it is 683:Was sind und was sollen die Zahlen 50: 38: 14: 6567: 5674: 4912:The American Mathematical Monthly 3135:For example: "A general function 2999:N. Bourbaki (18 September 2003). 2963:"History of the function concept" 2071:by means of the five operations . 1864:Independence (mathematical logic) 1701:is the dependent variable, e.g., 717:at any rate correspond values of 644:Similarly, Lavine observes that: 417:Théorie Analytique de la Chaleur, 259:introduced the familiar notation 114:Functions before the 17th century 94:and others, the reformulation of 6515: 6514: 6100: 5312:Incomplete symbols: Descriptions 5071:(1995 ed.). Courier Dover. 4761:Théorie analytique de la chaleur 4529:. Hermann & cie. p. 76. 4515: 4490: 4461: 3881:Incomplete symbols: Descriptions 2530: 2172:) is different from the couple ( 1697:is the independent variable and 371:. Euler's own definition reads: 5500:The College Mathematics Journal 5396:An axiomatization of set theory 4852:Introduction to Metamathematics 4624:Mathematics-The Music of Reason 4479:, Oxford University Press, NY, 4452: 4439: 4422: 4411:, Oxford University Press, NY, 4393: 4377: 4368: 4351: 4334: 4314: 4295: 4267: 4247: 4227: 4211: 4194: 4154: 4125: 4108: 4092: 4075: 4059: 4042: 4025: 4006: 3987: 3971: 3958: 3915: 3902: 3870: 3857: 3848: 3839: 3761: 3750:All symbols used here are from 3744: 3700: 3684: 3670: 3654: 3637: 3620: 3604: 3583: 3558: 3541: 3493: 3464: 3451: 3438: 3377: 3364: 3325: 3294: 3229: 3213: 3197: 3184: 3171: 3147: 3129: 3115: 3043: 2978: 2373:An axiomatization of set theory 2323:". Whatever the case, by 1924, 1808:Gödel's incompleteness theorems 952:notions, e.g., 1 − 725:nor to associate each value of 596:also varies continuously, then 515:Lobachevsky (1834) writes that 5535:. Cambridge University Press. 5414:The foundations of mathematics 2950: 2872: 2859: 2104:The history of the notion of " 1781:The Foundations of Mathematics 582: 576: 535:while Dirichlet (1837) writes 506:Peter Gustav Lejeune Dirichlet 276: 270: 1: 5394:(1967) . "von Neumann (1925) 5368:(1967) . "Schönfinkel (1924) 5028:. Cambridge University Press. 4994:. Cambridge University Press. 4991:The Principles of Mathematics 4910:(1998). "Function: Part II". 4804:. New York, Springer-Verlag. 4691:10.1090/s0273-0979-07-01174-3 4572: 3863:Something similar appears in 2751:Relational form of a function 2624: 2511:) we write to indicate that 2383:At the outset he begins with 2359:Von Neumann's set theory 1925 2201:The Principles of Mathematics 1981:from a previously formed set 1353:The Principles of Mathematics 1344:The Principles of Mathematics 1113:(1913) jointly authored with 713:such that "to some values of 290:for the value of a function. 6546:Basic concepts in set theory 5438:Principia Mathematica to *56 4825:A Course of Pure Mathematics 4177:Whitehead & Russell 1913 4162:Whitehead & Russell 1913 3878:Whitehead & Russell 1910 3520:Axiomatization of Set Theory 1200:The Principles of Arithmetic 364:{\displaystyle {x^{2}+3x+2}} 255:(in approximately 1734) and 16:Abour mathematical functions 7: 4876:. Harvard University Press. 3514:, p. 151. See also in 3222:, p. 135 as quoted in 2152:) is the class of couples ( 1858:of the functional calculus 1806:'s effort was failure; see 1569:the propositional function 669: 527:and varying gradually with 474:arithmetization of analysis 10: 6572: 5618:Mathematical Intelligencer 5602:Elements of Symbolic Logic 5284:(1967) . "Zermelo (1908a) 5240:(1967) . "Russell (1908a) 4874:Understanding the Infinite 4830:Cambridge University Press 4477:Kurt Godel Collected Works 3279:may assume constitute the 3206:, p. 43 as quoted in 2638:"function" as a relation, 1862:is provable"; see more at 1852:number-theoretic functions 1482:in its raw form, e.g., as 1262:is an object of the class 1184:"... is lighter than ... " 777:structure of the statement 729:to a single value of 410: 6510: 6472: 6441: 6400: 6374: 6180: 6095: 6055: 6027: 6020: 5972:Necessity and sufficiency 5875: 5840: 5792: 5746: 5728: 5720: 5600:Reichenbach, Hans (1947) 5262:(1967) . "Zermelo (1908) 5218:(1967) . "Richard (1905) 5200:(1967) . "Hilbert (1904) 5164:(1967) . "Russell (1902) 4772:; Bornet, Gérard (1997). 3383:"By a mapping φ of a set 2833:10.1007/s10649-006-9023-7 2248:, 2}}. A few years later 1951:be predicated of itself?" 1898:. It was given a push by 1733:). Russell repeats that " 1409:is what is left when the 497:Lobachevsky and Dirichlet 451: 382:In 1755, however, in his 31:; for example, the slope 5565:10.1080/0020739800110404 5494:Kleiner, Israel (1989). 5412:(1967) . "Hilbert(1927) 5350:(1967) . "Skolem (1922) 5332:(1967) . "Wiener (1914) 5042:(1972 ed.). Dover. 4974:The Mathematics Educator 4637:Gesammelte Werke, Bd. I. 4632:Dirichlet, G. P. Lejeune 3518:the introduction to his 2973:University of St Andrews 2928:Kleiner, Israel (2009). 2806: 2589:, there exists a unique 1749:" would be written " sin 1306:is the function presign 1008:, and "the special law" 928:is termed a function of 697: 589:{\displaystyle {y=f(x)}} 308: 295:differentiable functions 5531:. In Roy Porter (ed.). 5527:Lützen, Jesper (2003). 5429:Whitehead, Alfred North 5304:Whitehead, Alfred North 5182:(1967) . "Frege (1902) 5146:(1967) . "Peano (1889) 5128:(1967) . "Frege (1879) 5006:(2nd ed.). Dover. 4740:(3rd ed.). Dover. 4409:From Brouwer to Hilbert 3322:Published posthumously. 2936:. MAA. pp. 14–26. 2771:"An expression such as 2569:and a variable element 1518:—i.e., if its truth is 1475:Propositional functions 1434:and an assertion about 1389:propositional functions 1128:", replacing them with 539:If now a unique finite 6551:History of mathematics 6541:Functions and mappings 6433:Medieval Islamic world 6169:History of mathematics 5056:Chapter 1 Introduction 4770:Grattan-Guinness, Ivor 4602:. Walton and Marberly. 4588:. Walton and Marberly. 3307:Proofs and Refutations 2550:, gave the following: 2217:Alfred North Whitehead 2209:axioms of reducibility 1930:Russell's paradox 1902 1480:propositional function 1115:Alfred North Whitehead 824:and that of horses by 590: 543:corresponding to each 365: 284: 283:{\displaystyle {f(x)}} 253:Alexis Claude Clairaut 189:infinitesimal calculus 122:analyzed the equation 61: 6502:Future of mathematics 6479:Women in mathematics 6107:Philosophy portal 4820:Hardy, Godfrey Harold 4204:Principia Mathematica 3643:This example is from 3555:= 0 or 1, on page 37. 3332:Gardiner, A. (1982). 3259:is a (single-valued) 2739:, formally defines a 2297:Principia Mathematica 2226:Axiom of reducibility 2221:Principia Mathematica 2164:"Such a couple has a 2130:Principia Mathematica 1924:axiom of reducibility 1832:μ-recursive functions 1456:Principia Mathematica 1452:Principia Mathematica 1282:are objects of class 1274:; secondly, that "if 1110:Principia Mathematica 735:multivalued functions 612:be given in terms of 591: 366: 285: 120:Sharaf al-Din al-Tusi 62: 6454:Over Cantor's theory 5101:van Heijenoort, Jean 5039:Axiomatic Set Theory 4882:Lobachevsky, Nikolai 4848:Kleene, Stephen Cole 4207:. 1913. p. 161. 3845:Russell 1910–1913:15 3281:domain of definition 3269:independent variable 3267:. . . is called the 3194:, pp. 135–160). 2959:Robertson, Edmund F. 2737:Axiomatic Set Theory 2671:) (in Suppes) and S( 2654:Axiomatic Set Theory 2649:While the reader of 2168:, i.e., the couple ( 1916:Burali-Forti paradox 1868:Computability theory 1844:computable functions 1812:Entscheidungsproblem 1674:(pages 30ff) as the 1672:descriptive function 1630:: "Bob is hurt" and 1399:Function und Begriff 894:axiomatic set theory 828:and the conjunction 739:computability theory 563: 331: 263: 35: 6490:Approximations of π 6401:By ancient cultures 5769:Philosophy of logic 5420:. pp. 464–479. 5402:. pp. 393–413. 5378:With commentary by 5376:. pp. 355–366. 5358:. pp. 290–301. 5340:. pp. 224–227. 5320:With commentary by 5318:. pp. 216–223. 5296:definite properties 5292:. pp. 199–215. 5270:. pp. 183–198. 5250:With commentary by 5248:. pp. 150–182. 5226:. pp. 142–144. 5208:. pp. 129–138. 5190:. pp. 126–128. 5172:. pp. 124–125. 4940:Medvedev, Fyodor A. 4888:. Moscow-Leningrad. 4778:. Springer-Verlag. 4594:De Morgan, Augustus 4512:, pp. 143–145. 4447:van Heijenoort 1967 4434:van Heijenoort 1967 4388:van Heijenoort 1967 4365:, pp. 394–396. 4363:van Heijenoort 1967 4346:van Heijenoort 1967 4342:Curry and Feys 1958 4329:van Heijenoort 1967 4309:van Heijenoort 1967 4278:van Heijenoort 1967 4262:van Heijenoort 1967 4242:van Heijenoort 1967 4222:van Heijenoort 1967 4191:, pp. 523–529. 4137:van Heijenoort 1967 4120:van Heijenoort 1967 4103:van Heijenoort 1967 4087:van Heijenoort 1967 4070:van Heijenoort 1967 4054:van Heijenoort 1967 4037:van Heijenoort 1967 4020:van Heijenoort 1967 4001:van Heijenoort 1967 3982:van Heijenoort 1967 3910:van Heijenoort 1967 3885:van Heijenoort 1967 3756:van Heijenoort 1967 3739:van Heijenoort 1967 3724:van Heijenoort 1967 3712:van Heijenoort 1967 3695:van Heijenoort 1967 3665:van Heijenoort 1967 3649:van Heijenoort 1967 3632:van Heijenoort 1967 3615:van Heijenoort 1967 3599:van Heijenoort 1967 3595:van Heijenoort 1967 3524:van Heijenoort 1967 3512:van Heijenoort 1967 3263:of x. The variable 2957:O'Connor, John J.; 2665:axiom of separation 2345:functional calculus 2205:The Theory of Types 2084:well-formed formula 2001:possesses a subset 1342:Bertrand Russell's 964:is the logical AND( 902:The Laws of Thought 817:The Nature of Logic 502:Nikolai Lobachevsky 178:Islamic mathematics 84:analytic expression 6293:Information theory 6068:Rules of inference 6037:Mathematical logic 5779:Semantics of logic 5659:10.1007/BF00348305 5630:10.1007/BF03026743 5586:10.1007/BF00348540 5366:Schönfinkel, Moses 4832:(published 1993). 4726:. Springer-Verlag. 4714:. Springer-Verlag. 4670:Edwards, Harold M. 4626:. Springer-Verlag. 4436:, pp. 396–398 4405:structural concept 4322:Schönfinkel (1924) 4302:Schönfinkel (1924) 4105:, pp. 292–293 3887:, pp. 216–223 3181:, pp. 31–80). 2763:, is explained by 2431:) whose variables 2283:Where exactly the 2088:predicate calculus 1971:) to "separate" a 1466:or "the father of 1266:, the expression φ 980:is the logical OR( 904:1854; John Venn's 762:Augustus De Morgan 586: 490:Analyse algébrique 361: 280: 57: 6528: 6527: 6364:Separation axioms 6135: 6134: 6091: 6090: 5925:Deductive closure 5871: 5870: 5810:Critical thinking 5448:978-0-521-62606-4 5433:Russell, Bertrand 5392:von Neumann, John 5384:combinatory logic 5308:Russell, Bertrand 5238:Russell, Bertrand 5184:Letter to Russell 5162:Russell, Bertrand 5154:. pp. 83–97. 5000:Russell, Bertrand 4986:Russell, Bertrand 4966:Ponte, João Pedro 4861:978-0-7204-2103-3 4839:978-0-521-09227-2 4608:Dedekind, Richard 4399:In his 1930–1931 4160:both quotes from 4016:Letter to Russell 3939:978-0-486-43228-1 3921:Turing 1936–7 in 3634:, pp. 21–22. 3580:, pp. 86–87. 3470:Boole circa 1849 3271:and the variable 3226:, pp. 60–61. 3012:978-3-540-65340-0 2943:978-0-88385-569-0 2882:, pp. 29–30) 2541:cartesian product 2411:for the argument 2353:combinatory logic 2325:Moses Schönfinkel 2264:}}". As noted by 2128:By 1910–1913 and 1912:Russell's paradox 1737:is a function of 1731:x R y 1454:. By the time of 775:depends upon the 743:partial functions 229:Gottfried Leibniz 185:analytic geometry 6563: 6556:History of logic 6518: 6517: 6238:Category theory 6162: 6155: 6148: 6139: 6138: 6105: 6104: 6103: 6025: 6024: 5790: 5789: 5754:Computer science 5715: 5708: 5701: 5692: 5691: 5670: 5641: 5597: 5568: 5546: 5523: 5490: 5478: 5452: 5421: 5408:——; 5403: 5390:——; 5377: 5364:——; 5359: 5346:——; 5341: 5328:——; 5319: 5302:——; 5293: 5280:——; 5271: 5258:——; 5249: 5236:——; 5227: 5214:——; 5209: 5196:——; 5191: 5178:——; 5173: 5160:——; 5155: 5142:——; 5137: 5136:. pp. 1–82. 5124:——; 5120: 5096: 5082: 5053: 5029: 5017: 4995: 4981: 4961: 4949: 4935: 4903: 4897: 4889: 4877: 4870:Lavine, Shaughan 4865: 4843: 4815: 4803: 4800:Naive Set Theory 4789: 4765: 4751: 4727: 4715: 4703: 4693: 4665: 4659: 4651: 4627: 4615: 4603: 4589: 4567: 4561: 4555: 4549: 4543: 4537: 4531: 4530: 4519: 4513: 4507: 4501: 4496:All quotes from 4494: 4488: 4465: 4459: 4456: 4450: 4443: 4437: 4430:von Neumann 1925 4428:All quotes from 4426: 4420: 4397: 4391: 4384:von Neumann 1925 4381: 4375: 4372: 4366: 4359:von Neumann 1925 4355: 4349: 4338: 4332: 4318: 4312: 4299: 4293: 4287: 4281: 4271: 4265: 4251: 4245: 4231: 4225: 4215: 4209: 4208: 4198: 4192: 4186: 4180: 4174: 4165: 4158: 4152: 4146: 4140: 4129: 4123: 4112: 4106: 4096: 4090: 4079: 4073: 4063: 4057: 4046: 4040: 4029: 4023: 4010: 4004: 3991: 3985: 3975: 3969: 3962: 3956: 3950: 3944: 3943: 3919: 3913: 3906: 3900: 3894: 3888: 3874: 3868: 3861: 3855: 3852: 3846: 3843: 3837: 3831: 3825: 3819: 3813: 3807: 3801: 3795: 3784: 3765: 3759: 3748: 3742: 3736: 3727: 3721: 3715: 3704: 3698: 3697:, pp. 21–24 3688: 3682: 3674: 3668: 3667:, pp. 21–22 3658: 3652: 3651:, pp. 21–22 3641: 3635: 3624: 3618: 3608: 3602: 3587: 3581: 3575: 3569: 3562: 3556: 3545: 3539: 3533: 3527: 3516:von Neumann 1925 3497: 3491: 3485: 3479: 3468: 3462: 3455: 3449: 3442: 3436: 3430: 3424: 3418: 3412: 3381: 3375: 3368: 3362: 3356: 3350: 3349: 3329: 3323: 3321: 3298: 3292: 3233: 3227: 3217: 3211: 3204:Lobachevsky 1951 3201: 3195: 3188: 3182: 3179:Lobachevsky 1951 3175: 3169: 3163: 3157: 3151: 3145: 3133: 3127: 3119: 3113: 3107: 3101: 3095: 3089: 3083: 3077: 3071: 3065: 3059: 3053: 3047: 3041: 3035: 3029: 3023: 3017: 3016: 2996: 2990: 2982: 2976: 2975: 2954: 2948: 2947: 2925: 2919: 2913: 2907: 2901: 2895: 2889: 2883: 2876: 2870: 2863: 2857: 2851: 2845: 2844: 2816: 2727:functional graph 2715:functional graph 2661:Naive Set Theory 2365:Abraham Fraenkel 2341:Schönfinkel 1924 1908:Cantor's paradox 1880:De Morgan (1847) 1596:Russell defines 1171:Frege calls the 1095:(1879) preceded 944:Boole then used 595: 593: 592: 587: 585: 370: 368: 367: 362: 360: 344: 343: 289: 287: 286: 281: 279: 245:Johann Bernoulli 206: 175: 171: 167: 162: 158: 145: ⋅ ( 139: 66: 64: 63: 58: 49: 6571: 6570: 6566: 6565: 6564: 6562: 6561: 6560: 6531: 6530: 6529: 6524: 6506: 6468: 6449:Brouwer–Hilbert 6437: 6396: 6375:Numeral systems 6370: 6232:Grandi's series 6176: 6166: 6136: 6131: 6101: 6099: 6087: 6051: 6042:Boolean algebra 6016: 5867: 5858:Metamathematics 5836: 5788: 5742: 5724: 5719: 5677: 5543: 5512:10.2307/2686848 5481: 5475: 5459: 5457:Further reading 5449: 5382:. The start of 5348:Skolem, Thoralf 5330:Wiener, Norbert 5230:Richard paradox 5166:Letter to Frege 5144:Peano, Giuseppe 5117: 5079: 5050: 5034:Suppes, Patrick 5022:Smithies, Frank 5014: 4958: 4924:10.2307/2589085 4891: 4890: 4862: 4840: 4812: 4786: 4756:Fourier, Joseph 4748: 4720:Euler, Leonhard 4708:Euler, Leonhard 4653: 4652: 4648: 4620:Dieudonné, Jean 4575: 4570: 4562: 4558: 4550: 4546: 4538: 4534: 4520: 4516: 4508: 4504: 4495: 4491: 4466: 4462: 4457: 4453: 4444: 4440: 4427: 4423: 4398: 4394: 4382: 4378: 4373: 4369: 4356: 4352: 4339: 4335: 4319: 4315: 4300: 4296: 4288: 4284: 4272: 4268: 4252: 4248: 4232: 4228: 4216: 4212: 4200: 4199: 4195: 4187: 4183: 4175: 4168: 4159: 4155: 4147: 4143: 4130: 4126: 4113: 4109: 4097: 4093: 4080: 4076: 4064: 4060: 4047: 4043: 4033:Letter to Frege 4030: 4026: 4011: 4007: 3997:Letter to Frege 3992: 3988: 3976: 3972: 3963: 3959: 3951: 3947: 3940: 3920: 3916: 3907: 3903: 3895: 3891: 3875: 3871: 3862: 3858: 3853: 3849: 3844: 3840: 3832: 3828: 3820: 3816: 3812:, pp. 5–6. 3808: 3804: 3796: 3787: 3773:Laws of Thought 3766: 3762: 3749: 3745: 3737: 3730: 3722: 3718: 3708:Begriffsschrift 3705: 3701: 3689: 3685: 3675: 3671: 3659: 3655: 3642: 3638: 3625: 3621: 3609: 3605: 3588: 3584: 3576: 3572: 3563: 3559: 3546: 3542: 3534: 3530: 3498: 3494: 3486: 3482: 3469: 3465: 3456: 3452: 3448:, pp. 1, 2 3443: 3439: 3431: 3427: 3419: 3415: 3403:, denoted as φ( 3382: 3378: 3369: 3365: 3357: 3353: 3346: 3330: 3326: 3318: 3299: 3295: 3285:range of values 3234: 3230: 3218: 3214: 3202: 3198: 3189: 3185: 3176: 3172: 3164: 3160: 3152: 3148: 3134: 3130: 3120: 3116: 3108: 3104: 3096: 3092: 3084: 3080: 3072: 3068: 3060: 3056: 3048: 3044: 3036: 3032: 3024: 3020: 3013: 2997: 2993: 2989:, p. 234). 2983: 2979: 2955: 2951: 2944: 2926: 2922: 2914: 2910: 2902: 2898: 2890: 2886: 2877: 2873: 2864: 2860: 2852: 2848: 2817: 2813: 2809: 2779: + 2773:x is an integer 2753: 2632: 2627: 2533: 2502: 2495: 2488: 2481: 2474: 2467: 2460: 2453: 2361: 2306: 2281: 2180: = 2156:) for which φ( 2102: 2024:As there is no 2007: 1980: 1965: 1936:Begriffsschrift 1932: 1920:Richard paradox 1876: 1848:Church's thesis 1840:Turing machines 1824:lambda calculus 1794:choice function 1767: 1724: 1659: 1648: 1617: 1602:truth-functions 1589:" for variable 1444: 1394:Begriffsschrift 1362:Begriffsschrift 1348: 1204: 1093:Begriffsschrift 1086: 1082:Begriffsschrift 1004: + 996: + 976: + 956:is logical NOT( 910: 900:George Boole's 859: 836: + 807: 751: 700: 672: 566: 564: 561: 560: 499: 482:complex numbers 468:rather than on 454: 442:observes that: 413: 339: 335: 334: 332: 329: 328: 311: 266: 264: 261: 260: 225: 204: 173: 169: 165: 160: 141: 135: ⋅ 123: 116: 45: 36: 33: 32: 17: 12: 11: 5: 6569: 6559: 6558: 6553: 6548: 6543: 6526: 6525: 6523: 6522: 6511: 6508: 6507: 6505: 6504: 6499: 6498: 6497: 6487: 6486: 6485: 6476: 6474: 6470: 6469: 6467: 6466: 6461: 6459:Leibniz–Newton 6456: 6451: 6445: 6443: 6439: 6438: 6436: 6435: 6430: 6425: 6420: 6418:Ancient Greece 6415: 6410: 6404: 6402: 6398: 6397: 6395: 6394: 6389: 6384: 6378: 6376: 6372: 6371: 6369: 6368: 6367: 6366: 6361: 6360: 6359: 6346: 6345: 6344: 6339: 6329: 6328: 6327: 6321:Number theory 6319: 6314: 6313: 6312: 6302: 6301: 6300: 6290: 6285: 6284: 6283: 6278: 6268: 6267: 6266: 6256: 6251: 6250: 6249: 6244: 6236: 6235: 6234: 6229: 6219: 6218: 6217: 6207: 6206: 6205: 6197: 6196: 6195: 6184: 6182: 6178: 6177: 6165: 6164: 6157: 6150: 6142: 6133: 6132: 6130: 6129: 6124: 6114: 6109: 6096: 6093: 6092: 6089: 6088: 6086: 6085: 6080: 6075: 6070: 6065: 6059: 6057: 6053: 6052: 6050: 6049: 6044: 6039: 6033: 6031: 6022: 6018: 6017: 6015: 6014: 6009: 6004: 5999: 5994: 5989: 5984: 5979: 5974: 5969: 5964: 5959: 5954: 5949: 5948: 5947: 5937: 5932: 5927: 5922: 5917: 5916: 5915: 5910: 5900: 5895: 5890: 5885: 5879: 5877: 5873: 5872: 5869: 5868: 5866: 5865: 5860: 5855: 5850: 5844: 5842: 5838: 5837: 5835: 5834: 5829: 5824: 5819: 5818: 5817: 5812: 5802: 5796: 5794: 5787: 5786: 5781: 5776: 5771: 5766: 5761: 5756: 5750: 5748: 5744: 5743: 5741: 5740: 5735: 5729: 5726: 5725: 5718: 5717: 5710: 5703: 5695: 5689: 5688: 5676: 5675:External links 5673: 5672: 5671: 5642: 5613: 5598: 5569: 5559:(4): 489–492. 5548: 5541: 5524: 5491: 5483:Frege, Gottlob 5479: 5473: 5458: 5455: 5454: 5453: 5447: 5425: 5424: 5423: 5410:Hilbert, David 5405: 5387: 5361: 5343: 5325: 5299: 5282:Zermelo, Ernst 5277: 5260:Zermelo, Ernst 5255: 5233: 5216:Richard, Jules 5211: 5198:Hilbert, David 5193: 5180:Frege, Gottlob 5175: 5157: 5139: 5126:Frege, Gottlob 5115: 5097: 5092:Symbolic Logic 5083: 5077: 5063:Tarski, Alfred 5059: 5048: 5030: 5018: 5012: 4996: 4982: 4962: 4956: 4950:. Birkhauser. 4936: 4918:(3): 263–270. 4904: 4878: 4866: 4860: 4844: 4838: 4816: 4810: 4790: 4784: 4766: 4752: 4746: 4728: 4716: 4704: 4684:(4): 575–580. 4666: 4646: 4628: 4616: 4604: 4590: 4574: 4571: 4569: 4568: 4566:, p. 102. 4556: 4544: 4532: 4514: 4502: 4489: 4460: 4451: 4449:, p. 397. 4438: 4421: 4392: 4376: 4367: 4350: 4348:, p. 357. 4333: 4331:, p. 356. 4313: 4294: 4282: 4266: 4264:, p. 224. 4246: 4244:, p. 224. 4226: 4210: 4193: 4181: 4166: 4153: 4141: 4124: 4122:, p. 285. 4107: 4091: 4074: 4058: 4041: 4024: 4005: 3994:Russell (1902) 3986: 3970: 3968:, p. 268. 3957: 3945: 3938: 3914: 3901: 3889: 3869: 3856: 3847: 3838: 3826: 3814: 3802: 3800:, p. 505. 3785: 3777:Ernst Schröder 3760: 3758:, p. 91). 3743: 3728: 3716: 3699: 3683: 3681:observes this. 3669: 3653: 3636: 3619: 3603: 3601:, p. 86). 3582: 3570: 3557: 3540: 3528: 3492: 3490:, p. 222. 3480: 3463: 3457:Boole 1848 in 3450: 3444:Boole 1848 in 3437: 3433:De Morgan 1847 3425: 3423:, p. 135. 3421:Dieudonné 1992 3413: 3376: 3363: 3351: 3344: 3324: 3316: 3293: 3255:, then we say 3228: 3220:Dirichlet 1889 3212: 3196: 3192:Dirichlet 1889 3183: 3170: 3168:, p. 187. 3158: 3146: 3144:, p. 552) 3128: 3114: 3102: 3090: 3078: 3066: 3054: 3042: 3040:, p. 235. 3030: 3028:, p. 234. 3018: 3011: 2991: 2977: 2949: 2942: 2920: 2918:, p. 256. 2908: 2906:, p. 255. 2896: 2884: 2871: 2858: 2854:Dieudonné 1992 2846: 2810: 2808: 2805: 2797: 2796: 2785: 2784: 2752: 2749: 2631: 2628: 2626: 2623: 2532: 2529: 2500: 2493: 2486: 2479: 2472: 2465: 2458: 2451: 2391:, two objects 2381: 2380: 2369:Thoralf Skolem 2360: 2357: 2333: 2332: 2304: 2280: 2277: 2215:form); he and 2190: 2189: 2186:ordered couple 2146:ordered couple 2126: 2125: 2118:Russell (1903) 2110:Norbert Wiener 2101: 2098: 2093: 2092: 2086:in the simple 2076:van Heijenoort 2073: 2072: 2022: 2021: 2005: 1978: 1964: 1961: 1953: 1952: 1931: 1928: 1875: 1872: 1828:Stephen Kleene 1789: 1788: 1766: 1763: 1722: 1711: 1710: 1657: 1646: 1615: 1443: 1440: 1347: 1340: 1203: 1196: 1195: 1194: 1186: 1185: 1169: 1168: 1157: 1156: 1153: 1150: 1105:Russell (1903) 1097:Giuseppe Peano 1085: 1078: 1025:Symbolic Logic 942: 941: 909: 906:Symbolic Logic 898: 858: 855: 854: 853: 813: 812: 809: 805: 750: 747: 699: 696: 695: 694: 671: 668: 659: 658: 638: 637: 634:has two values 618: 617: 584: 581: 578: 575: 572: 569: 533: 532: 498: 495: 494: 493: 453: 450: 449: 448: 424:Fourier series 412: 409: 408: 407: 399: 398: 377: 376: 359: 356: 353: 350: 347: 342: 338: 310: 307: 278: 275: 272: 269: 257:Leonhard Euler 224: 221: 209: 208: 115: 112: 56: 52: 48: 44: 40: 15: 9: 6: 4: 3: 2: 6568: 6557: 6554: 6552: 6549: 6547: 6544: 6542: 6539: 6538: 6536: 6521: 6513: 6512: 6509: 6503: 6500: 6496: 6493: 6492: 6491: 6488: 6484: 6481: 6480: 6478: 6477: 6475: 6471: 6465: 6464:Hobbes–Wallis 6462: 6460: 6457: 6455: 6452: 6450: 6447: 6446: 6444: 6442:Controversies 6440: 6434: 6431: 6429: 6426: 6424: 6421: 6419: 6416: 6414: 6413:Ancient Egypt 6411: 6409: 6406: 6405: 6403: 6399: 6393: 6390: 6388: 6385: 6383: 6380: 6379: 6377: 6373: 6365: 6362: 6358: 6355: 6354: 6353: 6350: 6349: 6347: 6343: 6340: 6338: 6335: 6334: 6333: 6330: 6326: 6323: 6322: 6320: 6318: 6317:Math notation 6315: 6311: 6308: 6307: 6306: 6303: 6299: 6296: 6295: 6294: 6291: 6289: 6286: 6282: 6279: 6277: 6274: 6273: 6272: 6269: 6265: 6262: 6261: 6260: 6257: 6255: 6254:Combinatorics 6252: 6248: 6245: 6243: 6240: 6239: 6237: 6233: 6230: 6228: 6225: 6224: 6223: 6220: 6216: 6213: 6212: 6211: 6208: 6204: 6201: 6200: 6198: 6194: 6191: 6190: 6189: 6186: 6185: 6183: 6179: 6174: 6170: 6163: 6158: 6156: 6151: 6149: 6144: 6143: 6140: 6128: 6125: 6122: 6118: 6115: 6113: 6110: 6108: 6098: 6097: 6094: 6084: 6083:Logic symbols 6081: 6079: 6076: 6074: 6071: 6069: 6066: 6064: 6061: 6060: 6058: 6054: 6048: 6045: 6043: 6040: 6038: 6035: 6034: 6032: 6030: 6026: 6023: 6019: 6013: 6010: 6008: 6005: 6003: 6000: 5998: 5995: 5993: 5990: 5988: 5985: 5983: 5980: 5978: 5975: 5973: 5970: 5968: 5965: 5963: 5962:Logical truth 5960: 5958: 5955: 5953: 5950: 5946: 5943: 5942: 5941: 5938: 5936: 5933: 5931: 5928: 5926: 5923: 5921: 5918: 5914: 5911: 5909: 5906: 5905: 5904: 5903:Contradiction 5901: 5899: 5896: 5894: 5891: 5889: 5886: 5884: 5881: 5880: 5878: 5874: 5864: 5861: 5859: 5856: 5854: 5851: 5849: 5848:Argumentation 5846: 5845: 5843: 5839: 5833: 5832:Philosophical 5830: 5828: 5827:Non-classical 5825: 5823: 5820: 5816: 5813: 5811: 5808: 5807: 5806: 5803: 5801: 5798: 5797: 5795: 5791: 5785: 5782: 5780: 5777: 5775: 5772: 5770: 5767: 5765: 5762: 5760: 5757: 5755: 5752: 5751: 5749: 5745: 5739: 5736: 5734: 5731: 5730: 5727: 5723: 5716: 5711: 5709: 5704: 5702: 5697: 5696: 5693: 5686: 5682: 5679: 5678: 5668: 5664: 5660: 5656: 5652: 5648: 5643: 5639: 5635: 5631: 5627: 5623: 5619: 5614: 5611: 5610:0-486-24004-5 5607: 5603: 5599: 5595: 5591: 5587: 5583: 5579: 5575: 5570: 5566: 5562: 5558: 5554: 5549: 5544: 5538: 5534: 5530: 5525: 5521: 5517: 5513: 5509: 5505: 5501: 5497: 5492: 5488: 5484: 5480: 5476: 5474:0-88385-081-8 5470: 5466: 5461: 5460: 5450: 5444: 5440: 5439: 5434: 5430: 5426: 5419: 5415: 5411: 5406: 5401: 5397: 5393: 5388: 5385: 5381: 5380:Willard Quine 5375: 5371: 5367: 5362: 5357: 5353: 5349: 5344: 5339: 5335: 5331: 5326: 5323: 5317: 5313: 5309: 5305: 5300: 5297: 5291: 5287: 5283: 5278: 5275: 5274:impredicative 5269: 5265: 5261: 5256: 5253: 5252:Willard Quine 5247: 5243: 5239: 5234: 5231: 5225: 5221: 5217: 5212: 5207: 5203: 5199: 5194: 5189: 5185: 5181: 5176: 5171: 5167: 5163: 5158: 5153: 5149: 5145: 5140: 5135: 5131: 5127: 5122: 5121: 5118: 5116:0-674-32449-8 5112: 5108: 5107: 5102: 5098: 5094: 5093: 5088: 5084: 5080: 5078:0-486-28462-X 5074: 5070: 5069: 5064: 5060: 5057: 5051: 5049:0-486-61630-4 5045: 5041: 5040: 5035: 5031: 5027: 5023: 5019: 5015: 5013:0-486-27724-0 5009: 5005: 5001: 4997: 4993: 4992: 4987: 4983: 4979: 4975: 4971: 4967: 4963: 4959: 4957:9780817625726 4953: 4948: 4947: 4941: 4937: 4933: 4929: 4925: 4921: 4917: 4913: 4909: 4905: 4901: 4895: 4887: 4883: 4879: 4875: 4871: 4867: 4863: 4857: 4853: 4849: 4845: 4841: 4835: 4831: 4827: 4826: 4821: 4817: 4813: 4811:9780387900926 4807: 4802: 4801: 4795: 4791: 4787: 4785:3-7643-5456-9 4781: 4777: 4776: 4771: 4767: 4763: 4762: 4757: 4753: 4749: 4747:0-486-69609-X 4743: 4739: 4738: 4733: 4729: 4725: 4721: 4717: 4713: 4709: 4705: 4701: 4697: 4692: 4687: 4683: 4679: 4675: 4671: 4667: 4663: 4657: 4649: 4647:9780828402255 4643: 4639: 4638: 4633: 4629: 4625: 4621: 4617: 4613: 4609: 4605: 4601: 4600: 4595: 4591: 4587: 4586: 4581: 4580:Boole, George 4577: 4576: 4565: 4560: 4554:, p. 98. 4553: 4548: 4541: 4536: 4528: 4524: 4518: 4511: 4506: 4499: 4493: 4486: 4485:0-19-514721-9 4482: 4478: 4474: 4470: 4464: 4455: 4448: 4442: 4435: 4431: 4425: 4418: 4417:0-19-509632-0 4414: 4410: 4406: 4402: 4396: 4390:, p. 396 4389: 4385: 4380: 4371: 4364: 4360: 4354: 4347: 4343: 4337: 4330: 4326: 4323: 4317: 4311:, p. 359 4310: 4306: 4303: 4298: 4292:, p. 46. 4291: 4286: 4280:, p. 224 4279: 4275: 4270: 4263: 4259: 4258:Wiener (1914) 4255: 4250: 4243: 4239: 4236: 4230: 4224:, p. 224 4223: 4219: 4214: 4206: 4205: 4197: 4190: 4185: 4179:, p. 26. 4178: 4173: 4171: 4163: 4157: 4151:, p. 99. 4150: 4145: 4139:, p. 226 4138: 4134: 4128: 4121: 4117: 4111: 4104: 4100: 4095: 4089:, p. 199 4088: 4084: 4078: 4072:, p. 203 4071: 4067: 4066:Zermelo 1908a 4062: 4056:, p. 202 4055: 4051: 4050:Zermelo 1908a 4045: 4039:, p. 124 4038: 4034: 4028: 4022:, p. 127 4021: 4017: 4014: 4009: 4003:, p. 124 4002: 3998: 3995: 3990: 3983: 3979: 3974: 3967: 3961: 3955:, p. 45. 3954: 3949: 3941: 3935: 3931: 3930: 3925: 3924:Davis, Martin 3918: 3912:, p. 466 3911: 3905: 3899:, p. 53. 3898: 3893: 3886: 3882: 3879: 3873: 3866: 3860: 3851: 3842: 3836:, p. 19. 3835: 3830: 3823: 3818: 3811: 3806: 3799: 3794: 3792: 3790: 3782: 3778: 3774: 3770: 3764: 3757: 3753: 3747: 3741:, p. 91. 3740: 3735: 3733: 3726:, p. 89. 3725: 3720: 3713: 3709: 3703: 3696: 3692: 3687: 3680: 3673: 3666: 3662: 3657: 3650: 3646: 3640: 3633: 3629: 3623: 3616: 3612: 3607: 3600: 3596: 3592: 3586: 3579: 3574: 3567: 3561: 3554: 3550: 3544: 3538:, p. 86. 3537: 3532: 3526:, p. 395 3525: 3521: 3517: 3513: 3509: 3506: 3505:Russell 1908a 3502: 3501:Willard Quine 3496: 3489: 3484: 3477: 3473: 3467: 3460: 3454: 3447: 3441: 3434: 3429: 3422: 3417: 3410: 3409:Dedekind 1995 3406: 3402: 3398: 3394: 3390: 3386: 3380: 3373: 3372:Medvedev 1991 3367: 3361:, p. 34. 3360: 3355: 3347: 3345:0-486-42538-X 3341: 3337: 3336: 3328: 3319: 3317:0-521-29038-4 3313: 3309: 3308: 3303: 3302:Lakatos, Imre 3297: 3291:, p. 235 3290: 3286: 3282: 3278: 3274: 3270: 3266: 3262: 3258: 3254: 3250: 3246: 3242: 3238: 3232: 3225: 3224:Medvedev 1991 3221: 3216: 3210:, p. 58. 3209: 3208:Medvedev 1991 3205: 3200: 3193: 3187: 3180: 3174: 3167: 3166:Smithies 1997 3162: 3155: 3150: 3143: 3138: 3132: 3125: 3118: 3111: 3106: 3100:, p. 47. 3099: 3094: 3088:, p. 47. 3087: 3086:Medvedev 1991 3082: 3076:, p. VI. 3075: 3070: 3063: 3058: 3052:, p. 235 3051: 3046: 3039: 3034: 3027: 3022: 3014: 3008: 3004: 3003: 2995: 2988: 2981: 2974: 2970: 2969: 2964: 2960: 2953: 2945: 2939: 2935: 2931: 2924: 2917: 2916:Gardiner 1982 2912: 2905: 2904:Gardiner 1982 2900: 2893: 2888: 2881: 2880:Medvedev 1991 2875: 2868: 2862: 2856:, p. 55. 2855: 2850: 2842: 2838: 2834: 2830: 2826: 2822: 2815: 2811: 2804: 2802: 2801:Alfred Tarski 2794: 2793: 2792: 2790: 2787:For his part 2782: 2778: 2774: 2770: 2769: 2768: 2766: 2765:Tarski (1946) 2762: 2761:Halmos (1970) 2758: 2757:Suppes (1960) 2748: 2746: 2742: 2738: 2734: 2733:Suppes (1960) 2730: 2728: 2724: 2720: 2716: 2712: 2708: 2704: 2700: 2696: 2691: 2689: 2684: 2683:Kleene (1952) 2680: 2678: 2674: 2670: 2666: 2662: 2659: 2658:Halmos (1970) 2655: 2652: 2651:Suppes (1960) 2647: 2645: 2641: 2637: 2622: 2620: 2616: 2612: 2608: 2604: 2600: 2596: 2592: 2588: 2584: 2580: 2576: 2572: 2568: 2564: 2560: 2556: 2551: 2549: 2545: 2542: 2538: 2531:Bourbaki 1939 2528: 2526: 2520: 2518: 2514: 2510: 2506: 2499: 2492: 2485: 2478: 2475:follow from ( 2471: 2464: 2457: 2450: 2446: 2442: 2438: 2434: 2430: 2426: 2422: 2418: 2414: 2410: 2406: 2402: 2398: 2394: 2390: 2386: 2378: 2377: 2376: 2374: 2370: 2366: 2356: 2354: 2350: 2349:Haskell Curry 2346: 2342: 2338: 2337:Willard Quine 2335:According to 2330: 2329: 2328: 2326: 2322: 2318: 2314: 2310: 2302: 2298: 2294: 2290: 2286: 2276: 2274: 2269: 2267: 2266:Suppes (1960) 2263: 2259: 2255: 2251: 2247: 2243: 2239: 2235: 2231: 2227: 2222: 2218: 2214: 2210: 2206: 2202: 2198: 2193: 2187: 2183: 2179: 2175: 2171: 2167: 2163: 2162: 2161: 2159: 2155: 2151: 2147: 2143: 2139: 2135: 2131: 2123: 2122: 2121: 2119: 2115: 2111: 2107: 2097: 2089: 2085: 2081: 2080: 2079: 2077: 2070: 2066: 2062: 2058: 2054: 2053: 2052: 2049: 2047: 2043: 2039: 2035: 2031: 2027: 2026:universal set 2019: 2015: 2011: 2004: 2000: 1996: 1992: 1988: 1987: 1986: 1984: 1977: 1974: 1970: 1960: 1957: 1950: 1946: 1942: 1941: 1940: 1937: 1927: 1925: 1921: 1918:and the 1905 1917: 1913: 1909: 1905: 1901: 1897: 1893: 1889: 1885: 1881: 1871: 1869: 1865: 1861: 1857: 1853: 1849: 1845: 1841: 1837: 1833: 1829: 1825: 1821: 1817: 1816:Alonzo Church 1813: 1809: 1805: 1801: 1797: 1795: 1786: 1785: 1784: 1782: 1779: 1775: 1771: 1770:David Hilbert 1762: 1760: 1756: 1752: 1748: 1744: 1740: 1736: 1732: 1728: 1720: 1716: 1708: 1704: 1700: 1696: 1692: 1689: 1688: 1687: 1685: 1681: 1677: 1673: 1669: 1665: 1663: 1656: 1652: 1645: 1641: 1637: 1633: 1629: 1625: 1621: 1614: 1610: 1606: 1603: 1599: 1594: 1592: 1588: 1584: 1580: 1576: 1572: 1568: 1565: 1562: 1558: 1554: 1550: 1544: 1541: 1537: 1533: 1529: 1525: 1521: 1517: 1513: 1509: 1505: 1501: 1497: 1493: 1489: 1485: 1481: 1477: 1476: 1471: 1469: 1465: 1461: 1457: 1453: 1449: 1439: 1437: 1433: 1429: 1425: 1421: 1416: 1412: 1408: 1405: + 1404: 1400: 1396: 1395: 1390: 1384: 1382: 1378: 1374: 1370: 1365: 1363: 1359: 1355: 1354: 1345: 1339: 1337: 1333: 1329: 1325: 1321: 1317: 1313: 1309: 1305: 1301: 1297: 1293: 1289: 1285: 1281: 1277: 1273: 1269: 1265: 1261: 1257: 1253: 1249: 1245: 1241: 1237: 1233: 1229: 1225: 1221: 1217: 1213: 1209: 1201: 1192: 1191: 1190: 1183: 1182: 1181: 1179: 1174: 1166: 1165: 1164: 1162: 1154: 1151: 1148: 1147: 1146: 1142: 1140: 1135: 1131: 1127: 1123: 1118: 1116: 1112: 1111: 1106: 1102: 1098: 1094: 1090: 1089:Gottlob Frege 1083: 1077: 1075: 1071: 1067: 1063: 1059: 1054: 1050: 1046: 1042: 1038: 1034: 1030: 1026: 1021: 1019: 1015: 1011: 1007: 1003: 999: 995: 991: 987: 983: 979: 975: 971: 967: 963: 959: 955: 951: 947: 939: 935: 931: 927: 923: 922: 921: 919: 915: 907: 903: 897: 895: 891: 887: 883: 878: 876: 872: 868: 864: 850: 846: 845: 844: 841: 839: 835: 831: 827: 823: 818: 810: 803: 802: 801: 798: 794: 790: 786: 782: 778: 774: 773:logical truth 769: 767: 763: 759: 755: 746: 744: 740: 736: 732: 728: 724: 720: 716: 712: 708: 704: 692: 688: 687: 686: 684: 680: 676: 667: 665: 656: 652: 647: 646: 645: 642: 635: 630: 629: 628: 626: 621: 615: 611: 607: 603: 599: 579: 573: 570: 567: 558: 554: 550: 546: 542: 538: 537: 536: 530: 526: 522: 518: 517: 516: 513: 511: 507: 503: 491: 487: 486: 485: 483: 479: 475: 471: 467: 463: 459: 445: 444: 443: 441: 437: 433: 432:wave equation 429: 425: 421: 418: 404: 403: 402: 396: 392: 391: 390: 388: 386: 380: 374: 373: 372: 357: 354: 351: 348: 345: 340: 336: 326: 322: 318: 317: 306: 304: 300: 296: 291: 273: 267: 258: 254: 250: 246: 242: 238: 234: 230: 220: 218: 214: 202: 198: 197: 196: 194: 190: 186: 181: 179: 156: 152: 148: 144: 138: 134: 130: 126: 121: 111: 109: 105: 101: 97: 93: 89: 85: 80: 78: 74: 70: 54: 46: 42: 30: 26: 23:concept of a 22: 6392:Hindu-Arabic 6288:Group theory 6276:Trigonometry 6258: 6247:Topos theory 6002:Substitution 5822:Mathematical 5747:Major fields 5685:cut-the-knot 5653:(1): 37–85. 5650: 5646: 5624:(4): 72–77. 5621: 5617: 5601: 5580:(1): 57–84. 5577: 5573: 5556: 5552: 5532: 5503: 5499: 5486: 5464: 5437: 5417: 5413: 5399: 5395: 5383: 5373: 5369: 5355: 5351: 5337: 5333: 5315: 5311: 5295: 5289: 5285: 5267: 5263: 5245: 5241: 5223: 5219: 5205: 5201: 5187: 5183: 5169: 5165: 5151: 5147: 5133: 5129: 5105: 5095:. Macmillan. 5091: 5067: 5055: 5038: 5025: 5003: 4990: 4977: 4973: 4945: 4915: 4911: 4885: 4873: 4851: 4824: 4799: 4794:Halmos, Paul 4774: 4760: 4736: 4732:Eves, Howard 4723: 4711: 4681: 4677: 4636: 4623: 4611: 4598: 4584: 4559: 4547: 4542:, p. 5. 4535: 4526: 4517: 4505: 4492: 4476: 4472: 4468: 4463: 4454: 4441: 4424: 4408: 4404: 4400: 4395: 4379: 4370: 4353: 4341: 4336: 4324: 4316: 4304: 4297: 4290:Russell 1920 4285: 4269: 4249: 4237: 4229: 4213: 4203: 4196: 4189:Russell 1903 4184: 4164:, p. 26 4156: 4149:Russell 1903 4144: 4127: 4115: 4110: 4094: 4082: 4077: 4061: 4044: 4032: 4027: 4015: 4013:Frege (1902) 4008: 3996: 3989: 3984:, p. 23 3973: 3960: 3948: 3928: 3917: 3904: 3892: 3880: 3872: 3859: 3850: 3841: 3834:Russell 1903 3829: 3824:, p. 7. 3822:Russell 1903 3817: 3810:Russell 1903 3805: 3798:Russell 1903 3783:, p. 10 3781:Russell 1903 3772: 3769:Russell 1903 3763: 3746: 3719: 3714:, p. 85 3707: 3702: 3686: 3679:Russell 1903 3672: 3656: 3639: 3622: 3606: 3585: 3573: 3560: 3552: 3543: 3531: 3519: 3507: 3495: 3483: 3478:, p. 40 3471: 3466: 3453: 3440: 3435:, p. 1. 3428: 3416: 3404: 3400: 3396: 3392: 3388: 3384: 3379: 3366: 3354: 3334: 3327: 3306: 3296: 3284: 3280: 3276: 3272: 3268: 3264: 3260: 3256: 3252: 3248: 3244: 3240: 3236: 3231: 3215: 3199: 3186: 3173: 3161: 3149: 3142:Fourier 1822 3136: 3131: 3117: 3110:Fourier 1822 3105: 3098:Edwards 2007 3093: 3081: 3069: 3064:, p. 3. 3057: 3045: 3033: 3021: 3001: 2994: 2980: 2966: 2952: 2933: 2923: 2911: 2899: 2887: 2874: 2861: 2849: 2824: 2820: 2814: 2798: 2786: 2780: 2776: 2772: 2754: 2744: 2740: 2736: 2731: 2726: 2722: 2718: 2714: 2710: 2706: 2702: 2698: 2694: 2692: 2687: 2681: 2676: 2672: 2668: 2664: 2660: 2653: 2648: 2643: 2639: 2635: 2633: 2618: 2614: 2610: 2609:the element 2606: 2602: 2598: 2594: 2590: 2586: 2582: 2581:if, for all 2578: 2574: 2570: 2566: 2562: 2558: 2554: 2552: 2547: 2543: 2534: 2521: 2516: 2515:, just like 2512: 2508: 2504: 2497: 2490: 2483: 2476: 2469: 2462: 2455: 2448: 2444: 2440: 2436: 2432: 2428: 2424: 2420: 2416: 2412: 2408: 2404: 2400: 2396: 2392: 2388: 2384: 2382: 2372: 2362: 2344: 2334: 2320: 2316: 2312: 2308: 2300: 2296: 2292: 2288: 2284: 2282: 2272: 2270: 2261: 2257: 2253: 2245: 2241: 2237: 2233: 2220: 2204: 2200: 2194: 2191: 2185: 2181: 2177: 2173: 2169: 2165: 2157: 2153: 2149: 2145: 2137: 2129: 2127: 2106:ordered pair 2103: 2094: 2078:summarizes: 2074: 2068: 2064: 2060: 2056: 2050: 2029: 2023: 2017: 2016:for which Φ( 2013: 2009: 2002: 1998: 1994: 1990: 1982: 1975: 1968: 1966: 1958: 1954: 1948: 1944: 1935: 1933: 1900:Georg Cantor 1896:Peano (1889) 1892:Frege (1879) 1877: 1859: 1855: 1799: 1798: 1790: 1780: 1778:Hilbert 1927 1768: 1758: 1754: 1750: 1746: 1742: 1738: 1734: 1730: 1726: 1718: 1714: 1712: 1706: 1702: 1698: 1694: 1690: 1683: 1679: 1675: 1671: 1667: 1666: 1661: 1654: 1650: 1643: 1639: 1635: 1631: 1627: 1623: 1619: 1612: 1608: 1604: 1601: 1597: 1595: 1590: 1586: 1582: 1578: 1574: 1570: 1566: 1563: 1560: 1556: 1552: 1548: 1547:exist, and " 1545: 1539: 1535: 1531: 1527: 1523: 1519: 1515: 1511: 1507: 1503: 1499: 1495: 1491: 1487: 1483: 1479: 1473: 1472: 1467: 1463: 1459: 1455: 1451: 1447: 1445: 1435: 1431: 1427: 1414: 1410: 1406: 1402: 1398: 1392: 1385: 1380: 1376: 1372: 1366: 1361: 1357: 1351: 1349: 1343: 1335: 1331: 1327: 1323: 1319: 1315: 1311: 1307: 1303: 1299: 1295: 1291: 1287: 1283: 1279: 1275: 1271: 1267: 1263: 1259: 1255: 1251: 1247: 1243: 1239: 1235: 1231: 1227: 1223: 1219: 1215: 1211: 1207: 1205: 1199: 1187: 1177: 1172: 1170: 1161:the function 1160: 1158: 1143: 1138: 1133: 1129: 1125: 1121: 1119: 1108: 1087: 1081: 1073: 1069: 1065: 1061: 1057: 1048: 1044: 1040: 1036: 1032: 1028: 1024: 1023:In his 1881 1022: 1017: 1013: 1009: 1005: 1001: 997: 993: 989: 985: 981: 977: 973: 969: 965: 961: 957: 953: 949: 945: 943: 937: 933: 929: 925: 920:as follows: 917: 913: 911: 905: 901: 892:– the first 889: 882:Georg Cantor 879: 860: 848: 842: 837: 833: 829: 825: 821: 816: 815:In his 1848 814: 796: 792: 788: 784: 780: 776: 770: 766:George Boole 752: 737:and what in 730: 726: 722: 718: 714: 710: 706: 701: 690: 682: 681:in his work 673: 660: 654: 650: 643: 639: 633: 625:Imre Lakatos 622: 619: 613: 609: 605: 604:function of 601: 600:is called a 597: 556: 552: 548: 544: 540: 534: 528: 524: 520: 514: 500: 489: 455: 416: 414: 400: 394: 383: 381: 378: 314: 312: 292: 248: 235:, such as a 226: 216: 212: 210: 203:(1323–1382) 182: 154: 150: 146: 142: 140:in the form 136: 132: 128: 124: 117: 110:to another. 81: 72: 21:mathematical 18: 6408:Mesopotamia 6382:Prehistoric 6342:Probability 6199:Algorithms 6117:WikiProject 5987:Proposition 5982:Probability 5935:Description 5876:Foundations 5322:W. V. Quine 5276:definition. 4564:Tarski 1946 4552:Tarski 1946 4540:Tarski 1946 4510:Kleene 1952 4498:Suppes 1960 4487:(v.2, pbk). 4344:; Quine in 4274:Wiener 1914 4254:Suppes 1960 4235:Wiener 1914 4218:Wiener 1914 4133:Wiener 1914 4099:Skolem 1922 3953:Kleene 1952 3908:Hilbert in 3897:Kleene 1952 3865:Tarski 1946 3617:, p. 7 3568:, p. 6 3461:, p. 6 3411:, p. 9 3359:Lavine 1994 2725:instead of 2677:expressions 2367:(1922) and 2293:descriptive 2213:extensional 2134:intensional 2046:von Neumann 2020:) is true". 1888:Venn (1881) 1836:Alan Turing 1834:(1936) and 1743:descriptive 1557:significant 1524:truth-value 1520:determinate 1516:significant 1508:proposition 741:are called 462:Weierstrass 406:"function". 239:or curve's 193:Middle Ages 92:Weierstrass 6535:Categories 6332:Statistics 6264:Logarithms 6210:Arithmetic 6047:Set theory 5945:Linguistic 5940:Entailment 5930:Definition 5898:Consequent 5893:Antecedent 5542:0521571995 5087:Venn, John 4573:References 4523:N.Bourbaki 4469:relation ε 3978:Frege 1879 3752:Peano 1889 3691:Frege 1879 3661:Frege 1879 3645:Frege 1879 3628:Frege 1879 3611:Frege 1879 3591:Peano 1889 3549:Boole 1854 3536:Boole 1854 3503:preceding 3154:Luzin 1998 3124:Kolmogorov 3074:Euler 2000 3062:Euler 1988 2892:Ponte 1992 2867:Ponte 1992 2827:(2): 192. 2667:, e.g., φ( 2625:Since 1950 2389:II-objects 2250:Kuratowski 2197:antinomies 2114:Schröder's 1774:consistent 1420:predicates 1330:φ yields 1101:Frege 1879 852:Calculus." 797:predicate. 758:syllogisms 703:Hardy 1908 655:definition 651:continuous 602:continuous 466:arithmetic 436:d'Alembert 428:continuous 321:expression 237:coordinate 100:set theory 6352:Manifolds 6348:Topology 6259:Functions 6078:Fallacies 6073:Paradoxes 6063:Logicians 5997:Statement 5992:Reference 5957:Induction 5920:Deduction 5883:Abduction 5853:Metalogic 5800:Classical 5764:Inference 5681:Functions 5667:121038818 5638:189883712 5594:120506760 5103:(1976) . 4980:(2): 3–8. 4908:Luzin, N. 4894:cite book 4656:cite book 3966:Eves 1990 3578:Venn 1881 3566:Venn 1881 3488:Eves 1990 3289:Eves 1990 3050:Eves 1990 3038:Eves 1990 3026:Eves 1990 2987:Eves 1990 2841:120363574 2709:). Here 2535:In 1939, 2385:I-objects 2230:Hausdorff 2176:) unless 2142:extension 2138:extension 1820:algorithm 1755: 'y 1751: 'y 1567:satisfies 1506:yields a 1462:" or log 1424:relations 1373:variables 1310:+, then φ 1250:)) THEN ( 1126:predicate 946:algebraic 871:Logicists 754:Logicians 675:Dieudonné 395:functions 6520:Category 6495:timeline 6483:timeline 6357:timeline 6337:timeline 6325:timeline 6310:timeline 6298:timeline 6281:timeline 6271:Geometry 6242:timeline 6227:timeline 6222:Calculus 6215:timeline 6203:timeline 6193:timeline 6181:By topic 6173:timeline 6112:Category 6012:Validity 5913:Antinomy 5841:Theories 5805:Informal 5489:. Halle. 5485:(1879). 5435:(1913). 5089:(1881). 5065:(1946). 5054:cf. his 5036:(1960). 5024:(1997). 5002:(1920). 4988:(1903). 4968:(1992). 4942:(1991). 4884:(1951). 4872:(1994). 4850:(1952). 4822:(1908). 4796:(1970). 4758:(1822). 4734:(1990). 4722:(2000). 4710:(1988). 4672:(2007). 4640:Berlin. 4634:(1889). 4622:(1992). 4596:(1847). 4582:(1854). 4525:(1954). 3926:(1965). 3261:function 3237:variable 2745:function 2741:relation 2723:function 2719:op. cit. 2537:Bourbaki 2363:By 1925 2091:adopted. 2038:Fraenkel 2030:domain B 1904:antinomy 1886:(1880), 1826:(1936), 1561:variable 1369:variable 1358:function 1234:), IF (( 1198:Peano's 1173:argument 1163:, i.e., 1134:function 1130:argument 1080:Frege's 849:function 785:relation 781:function 691:function 679:Dedekind 670:Dedekind 664:Bourbaki 510:relation 470:geometry 303:calculus 187:and the 96:geometry 88:analysis 29:calculus 25:function 6387:Ancient 6188:Algebra 6127:changes 6119: ( 5977:Premise 5908:Paradox 5738:History 5733:Outline 5520:2686848 4932:2589085 4700:2338366 2285:general 2273:did not 2240:) as {{ 1804:Bernays 1757:has to 1618:: "NOT( 1536:satisfy 1532:falsity 1522:—has a 1314:yields 1286:and if 1242:) AND ( 1122:subject 950:logical 912:In his 886:Zermelo 867:classes 789:subject 420:Fourier 415:In his 411:Fourier 325:formula 6029:topics 5815:Reason 5793:Logics 5784:Syntax 5665: 5636: 5608: 5592: 5539: 5518: 5471: 5445: 5113: 5075: 5046: 5010: 4954: 4930: 4858: 4836: 4808: 4782: 4744: 4698: 4644: 4483: 4415: 3936: 3342: 3314: 3009: 2940: 2839: 2789:Tarski 2759:, and 2644:define 2640:define 2636:define 2244:,1}, { 2044:, and 2042:Skolem 1973:subset 1884:Jevons 1705:= sin( 1678:value 1676:unique 1622:) AND 1600:, and 795:, and 793:copula 458:Cauchy 452:Cauchy 299:limits 201:Oresme 104:Cantor 77:Oresme 6473:Other 6428:India 6423:China 6305:Logic 6056:other 6021:Lists 6007:Truth 5774:Proof 5722:Logic 5683:from 5663:S2CID 5634:S2CID 5590:S2CID 5516:JSTOR 4928:JSTOR 4886:Works 3397:image 2837:S2CID 2807:Notes 2713:is a 2553:"Let 2489:) = ( 2313:x R y 2166:sense 1540:class 1528:truth 1326:then 1226:) = ( 1208:class 1178:right 1000:) is 865:, or 698:Hardy 636:: ... 440:Luzin 309:Euler 241:slope 233:curve 205:. . . 69:graph 67:of a 6121:talk 5967:Name 5952:Form 5606:ISBN 5537:ISBN 5469:ISBN 5443:ISBN 5418:ibid 5400:ibid 5374:ibid 5356:ibid 5338:ibid 5316:ibid 5290:ibid 5268:ibid 5246:ibid 5224:ibid 5206:ibid 5188:ibid 5170:ibid 5152:ibid 5134:ibid 5111:ISBN 5073:ISBN 5044:ISBN 5008:ISBN 4952:ISBN 4900:link 4856:ISBN 4834:ISBN 4806:ISBN 4780:ISBN 4742:ISBN 4662:link 4642:ISBN 4481:ISBN 4413:ISBN 3934:ISBN 3775:and 3370:See 3340:ISBN 3312:ISBN 3243:and 3137:f(x) 3007:ISBN 2938:ISBN 2557:and 2461:and 2435:and 2395:and 2387:and 2301:R 'y 2260:}, { 2207:two 2174:y, x 2170:x, y 2158:x, y 2154:x, y 2150:x, y 2034:Weyl 1894:and 1866:and 1691:N.B. 1638:and 1422:and 1381:some 1346:1903 1278:and 1202:1889 1132:and 1124:and 1084:1879 1056:not- 1053:Venn 908:1881 875:1903 863:sets 709:and 504:and 478:real 215:and 153:) = 19:The 5863:Set 5655:doi 5626:doi 5582:doi 5561:doi 5508:doi 5416:". 5398:". 5372:". 5354:". 5336:". 5314:". 5288:". 5266:". 5244:". 5222:". 5204:". 5186:". 5168:". 5150:". 5132:". 4920:doi 4916:105 4686:doi 4432:in 4386:in 4361:in 4340:cf 4327:in 4307:in 4276:in 4260:in 4240:in 4220:in 4135:in 4118:in 4101:in 4085:in 4068:in 4052:in 4035:in 4018:in 3999:in 3980:in 3754:in 3693:in 3663:in 3647:in 3630:in 3613:in 3593:in 3547:cf 3522:in 3510:in 3474:in 3407:). 3399:of 3391:of 2829:doi 2735:in 2729:." 2697:= ( 2656:or 2573:of 2565:of 2355:". 2319:to 2305:DEF 2063:or 2012:of 1830:'s 1761:". 1735:R'y 1723:DEF 1719:R'y 1717:": 1660:): 1581:: " 1573:: " 1530:or 1526:of 1486:: " 1438:". 1379:or 1377:any 1298:= φ 1218:, ( 1091:'s 1039:), 988:), 972:), 960:), 840:". 830:and 804:" A 783:or 555:to 480:or 323:or 108:set 102:by 90:by 6537:: 5661:. 5651:16 5649:. 5632:. 5620:. 5588:. 5576:. 5557:11 5555:. 5514:. 5504:20 5502:. 5498:. 5431:; 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Index