Logic and Philosophy of Mathematics, Modern

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This article surveys many of the main positions that have been held in the logic and philosophy of mathematics from around 1800 up to recent times. Most attention is given to symbolic logics of some kind. No position has been definitive; indeed, especially over the last seventy years the variety has continually increased. To compensate for this article's lack of exhaustiveness, the bibliography is wide-ranging.

The Revival of Logic from the 1820s, and Its Algebraic Flourishing

Some major philosophers gave logic a high status in the seventeenth and early eighteenth centuries; in particular, Gottfried Wilhelm von Leibniz advocated it as a lingua characteristica, with an attendant calculus ratiocinator, while John Locke took it as a case of semeiotiké, the theory of signs. The subject did not flourish, however; thus, while René Descartes stressed rules for correct thinking and deduction, and Immanuel Kant viewed logic as analytic knowledge (with mathematics as synthetically a priori), neither philosopher gave logic itself much attention. Most figures in all disciplines were content to appeal to Aristotelian syllogistic logic, while mathematicians normally saw the Elements of Euclid as the apotheosis of rigor.

The revival in logic came from an unexpected quarter: the Elements of Logic (1826) by the English theologian Richard Whately. Within a decade four more editions of this treatise had appeared, and various contemporaries commented at length. The strength of the reception is puzzling, for Whately did not advocate any radical new stances; nevertheless, interest in logic increased considerably and some notable advances occurred, especially the quantification of the predicate (1827) of George Bentham, in which Aristotle's modes were greatly extended by admitting forms such as all As are some Bs.

These advances attracted the mathematician Augustus De Morgan, who from the 1840s symbolized in an algebraic manner the forms of syllogistic modes and the relationships between them. He also studied the logical form of the proofs in Euclid, and came away rather perplexed; for the flow of argument in Euclid involved far more than logical deductions. A more sweeping change to logic was effected by George Boole, who started out from a new algebraic principle. Take a "universe of discourse" (his phrase) 1 of, say, boxes, and divide it into the class x of black ones and the complementary class (1 x ) of nonblack ones. Then lay down the basic laws obeyed by x, including the novelty that x together with x is the same as x. Make deductions by formulating algebraically relationships between classes x, y, and using the laws and attendant theorems to find as a deduction the relationships between say, y and the other classes. Boole elaborated his method especially in The Laws of Thought (1854), where syllogisms occupied only the last chapter on logic. Finally, in 1860 De Morgan enriched syllogistics with a logic of (two-place) relations: at last the failure properly to handle, for example, John is older than Jack had been recognized.

The contributions of these two English algebraists were distinct: the American logician Charles S. Peirce conjoined them from the 1870s. One of his major insights, arrived at with his student O. H. Mitchell, was to individuate the "quantifier," both the existential there is an X such that and the universal for all X , and to stress the importance of watching quantifier order. Continuing the algebraic style, Peirce and Mitchell regarded these quantifiers as generalizations of logical disjunctions and conjunctions respectively. The interpretation of quantifiers was to be an enduring theme in the philosophy of logic. Their logic was extended, especially as an algebra, by the German Ernst Schröder in his vast Vorlesungen über die Algebra der Logik (18901905).

Many others wrote upon logic, especially in Britain; I note three figures. John Stuart Mill's System of Logic (1843, and many later editions), while not tied to syllogistics, relied much on it for an analysis of reasoning and deduction; mathematicization was absent, but in Mill's account of "induction," which we would regard as philosophy of science, he touched upon probability theory. Mill was broadly aligned to British empiricism: by great contrast, an influential "neo-Hegelianism" later became popular in academic circles. Francis H. Bradley's Principles of Logic (1883), its landmark, is an idealistic meditation upon the basic laws of logic (such as that of the excluded middle), judgments, and the reconciliation of thesis and antithesis in synthesis. Finally, Lewis Carroll produced some rather dull books on logic (1886; 1896); but in his Alice books (1865; 1871) he had brilliantly anticipated several concerns of those logicians of the next century who were to launch a new tradition that came to eclipse the algebraic logicians.

Set Theory and the Rise of Mathematical Logic

Especially from the 1820s with the Frenchman Augustin Louis Cauchy, mathematicians had become more sensitive to the need for rigor in proofs, carefully stating assumptions and definitions and formulating theorems in conditional form. Cauchy's approach was refined from the 1860s by the lectures at Berlin University of Karl Weierstrass. The main context was the calculus and its extension into mathematical analysis. Two consequences are of special import.

Axiomatized logics are far removed from the normal use of logic. In teaching and practice "natural deduction" has recently gained much popularity: state the "local" premises, and deduce conclusions from them using rules for introducing and eliminating connectives. The name was introduced in the 1930s by Hilbert's follower Gerhard Gentzen.

First, in the early 1870s the German Georg Cantor began to develop set theory, initially treating points and numbers but later as a general treatment of collections of things; he even claimed that sets enabled him to define the natural numbers, and by implication to reduce mathematics to sets. His theory differed from the traditional part-whole theory that dates back to the Greeks and that was used by, for example, the algebraic logicians; for Cantor distinguished membership of objects in a set from the inclusion of subsets of objects within it. More controversially, he also incorporated a mathematical theory of the "actual infinite," showing that infinities came in different sizes.

Second, from the 1880s the Italian mathematician Giuseppe Peano began to symbolize as much as possible not only the notions of mathematical analysis (including set theory) but also the logical connectives and predicates with quantification. Thereby he launched "mathematical logic" (his name, in the sense usually adopted today). With an impressive school of followers he came to handle a wide range of mathematical theories this way, and enchanted the young Englishman Bertrand Russell, who came across him in 1900. Russell quickly added a logic of relations to mathematical logic, and converted Cantor's claim that set theory could ground mathematics into the "logicist" thesis (a name provided later by Rudolf Carnap) that mathematical logic could ground sets and thereby (much) mathematics. Russell also found that the German Gottlob Frege had already asserted this thesis for arithmetic and some parts of mathematical analysis, though in a Platonist spirit very different from his own empiricism.

So much for the good news. The bad arrived soon afterwards, in the form of a paradox now named after Russell: a certain set was a member of itself if and only if it was not. Paradoxes are at least a nuisance in mathematical theories: when the theory in question encompasses logic itself, their presence is a disaster. Eventually Russell produced an articulation of logicism in Principia Mathematica (19101913), written with Alfred North Whitehead; but his and other paradoxes were avoided rather than solved, and by an unwieldy and epistemologically questionable "theory of types": the ensemble of objects was divided into individuals, sets of individuals, sets of sets of individuals, , ordered pairs of individuals, , and so on, and membership was severely restricted.

Axiomatics and the Rise of Proof and Model Theories

Meanwhile, contentment with Euclidian rigor was dissolving. From the 1860s non-Euclidean geometries had been accepted as legitimate theories, especially due to the insights of Bernhard Riemann. In addition, various mathematicians, including Peano, had noticed that Euclidian geometry itself needed several more axioms than Euclid had stated.

These developments made mathematicians still more aware of finding and expressing the assumptions involved in a theory. Another stimulus was the rise in importance of algebras (for example, among several, Boole's), each with its own basic laws. David Hilbert, the leading mathematician of his generation, studied geometries and algebras intensively, and was led around 1900 to seek for a mathematical means of studying axiom systems. He found it in "metamathematics" (his later name for it), in which a system was examined to establish its completeness, consistency, and independence.

One of Hilbert's axioms for geometry was a meta-assumption that the other axioms supplied all the objects required. This soon led the young American mathematician Oswald Veblen to consider two different sets of objects satisfying an axiom system; if the members of each could be put in one-one correspondence, then the system was "categorical." The study of (non-)categoricity enriched model theory considerably.

Set theory itself was influenced by these developments, for in 1908 Hilbert's follower Ernst Zermelo axiomatized it. His system included his own discovery, the "axiom of choice," a nonconstructive assumption that some colleagues found doubtful. A strong debate broke out; most participants agreed that the axiom was unavoidable. However, it was unacceptable for "constructivist" mathematicians, who admitted only procedures that built up mathematical objects in explicit stages: for them the axiom was outlawed, along with some standard proof methods such as by contradiction (to prove theorem T, assume not-T and get into a logical mess). The most prominent figure in this tradition was the Dutch mathematician L. E. J. Brouwer, who elaborated his alternative "intuitionistic" mathematics especially in the 1920s. At that time Hilbert's metamathematical program was in its definitive phase; but in 1927 Brouwer derided it as "formalism," a misleading name that Hilbert himself never used but that has regrettably become standard.

Gödel, Tarski, and the Individuation of Metatheory

More bad news for Hilbert arrived in 1931, when the young Austrian Kurt Gödel proved that his metamathematical program could not be achieved for arithmetic with quantification over integers, and thus a fortiori for almost all acclimatized mathematical theories T; for Gödel showed that, in order to prove the consistency of T, its metamathematics had to be richer in assumptions than T itself rather than poorer, as Hilbert had hoped. This theorem was a corollary of another one, which torpedoed logicism by showing that there was a proposition M statable within T but neither provable nor disprovable; thus T, assumed to be consistent, was undecidable.

In addition, Gödel's proof-method soon led to a focus on recursion and to computability as conceived by the Englishman Alan Turing, thus launching a link between logic and computing. Finally, for Gödel's proofs to work the distinction between T and its metatheory had to be observed rigidly, as a central feature of both metamathematics and metalogic. The Polish logician Alfred Tarski also stressed this distinction around this time, for formal systems in general; his motive was treating as metatheoretic the (meta-)proposition that a proposition was true if it corresponded to the facts.

With the re-creation of their country after World War I, the Poles produced a remarkable cohort of logicians, with Jan Łukasiewicz and Stanisłav Leśniewski as leaders, Luand Tarski perhaps the most dominant figure. While Hitler saw to its demise, several members emigrated; for example, Tarski was a major figure later in the development of model theory.

Logics and Pluralism from the 1940s

The consequences of Gödel's theorems were profound. For example, logicism was replaced, especially by the American W. V. Quine, by elaborate systems of set theory and logic such as those developed in Quine's Mathematical Logic (1940); however, Russellian reductions of the former to the latter were not claimed.

Quine adhered to classical logic, with its two truth-values. But nonclassical logics gained adherents from the 1930s. The American C. I. Lewis had been a pioneer already in the 1910s, when he advocated "modal logics," where necessity and possibility were adjoined in various ways to truth and falsehood, as a means of clarifying the undoubtedly messy treatment of implication in Principia Mathematica. But from the 1930s these logics were viewed as viable alternatives to classicism in their own right. Another notable adherent was Gödel's friend Carnap, especially with his Meaning and Necessity (1947).

Since then logics classical and nonclassical have proliferated enormously, with their metalogics; many connections have been forged with computer science and aspects of semantics and linguistics. Notable among many cases are deontic logic, with its focus on sentences using "ought" and applications to law; temporal logics, seeking means to formulate deductions such as John eats; thus John will have eaten; many-valued logics, allowing for values other than true and false; non-monotonic logic or defeasible reasoning, defined by the property (in the simplest case) of propositions A, B, and C that given A C, then A and B C does not hold; paraconsistent logics, where various heresies are entertained, such as paradoxical propositions being both true and false; and fuzzy logic and set theory, where the vagueness of John is very tall is itself studied mathematically, initially by engineers rather than professional logicians.

This last topic bears upon control theory, which became associated with the discipline of cybernetics during the 1950s; logics of some kind were used in areas such as machine learning and brain modeling. Such concerns then developed further within the field of artificial intelligence, and were involved also in topics such as the representation of knowledge, automated theorem proving, and the relationship between complexity and recursion. These and very many more topics are now also linked to theoretical computer science, where many (meta)logics are used, with further connections to programming, formal and natural languages, and linguistics.

Partly due to these connections a renaissance of interest has occurred in higher-order logic, where quantification is executed over predicates as well as individuals. This revises a practice followed freely by Frege and by Russell and Whitehead. Theories of types have also been revived, appearing in several of the above contexts.

Philosophies of Mathematics

Especially since World War II, new philosophies have emerged separate from logicism, formalism, and intuitionism. One kind is called naturalism, in which mathematical objects are said to be accessible to ordinary sense perception. Forms of Platonism are advocated, partly inspired by the enthusiasm of Gödel. Disaffection from concerns with mathematical truth has stimulated conventionalism; Henri Poincaré was an eminent early advocate. Versions of constructivist mathematics have been developed, broadly following the same prohibitions as Brouwer's but avoiding his peculiar philosophy; Erret Bishop was a notable figure. Structuralism emphasizes the ubiquity of structures, perhaps to excess; mathematics has them rather than is them.

Most of this philosophy is by and for philosophers; the mathematical content is usually limited to set theory and/or arithmetic, even among empiricists who claim to attend to mathematical practice. In particular, little is said about the creation and development of mathematical theories in the first place. Here the eminent Hungarian Georg Polya was an important contributor, with books such as Mathematics and Plausible Reasoning (1954) discussing themes such as proofs themselves modifying old theorems and motivating new ones. A rather neglected area is the philosophy of mechanics and (classical) mathematical physics, despite the attraction of physical interpretation as well as rigor and proof.

By contrast, quantum mechanics has gained much attention, partly in connection with the philosophy of probability, which has a rather separate history. What kind of knowledge is expressed by the probability that the next throw of the die will be 5 is 1/6 ? Influential answers include the logical (it is a deduction from axioms), propensity (it is an assertion about the die and also its environment), subjective or Bayesian (it is a rational belief, drawing upon prior performance), and frequency (it is the limiting case of evidence drawn from long runs of throws).

Concluding Remark

Whately would be astonished at the current range and variety of logics. However, there is often still a considerable professional distance between logicians and both mathematicians and philosophers; in the mid-1930s an Association of Symbolic Logic was created to provide a venue and a journal for those active in the field. Now perhaps too many logical and philosophical flowers are blooming, and many may wither for want of genuine need. One area of exploration is metametalogic and metametaphilosophy of mathematics, which have been largely ignored so far.

See also Logic ; Quantum .


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Aspray, William, and Philip Kitcher, eds. History and Philosophy of Modern Mathematics. Minneapolis: University of Minnesota Press, 1988.

Barwise, Jon, ed. Handbook of Mathematical Logic. Amsterdam and New York: North-Holland, 1977.

Benacerraf, Paul, and Hilary Putnam, eds. Philosophy of Mathematics: Selected Readings. 2nd. ed., Cambridge, U.K., and New York: Cambridge University Press, 1983.

Fraenkel, Abraham. Abstract Set Theory. Amsterdam: North-Holland, 1953. Outstanding bibliography.

Gabbay, Dov, and F. Guenthner, eds . Handbook of Philosophical Logic. 1st ed. 4 vols. Dordrecht and Boston: Reidel, 19831987. 2nd ed., much expanded, in progress. Dordrecht: Kluwer, 2000.

Grattan-Guinness, Ivor. The Search for Mathematical Roots, 18701940: Logics, Set Theories, and the Foundations of Mathematicsfrom Cantor through Russell to Gödel. Princeton: Princeton University Press, 2000.

Grattan-Guinness, Ivor, ed. Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. 2 vols. London: Routledge, 1994. Reprint, Baltimore: Johns Hopkins University Press, 2003. See especially Part 6.

Haack, Susan. Philosophy of Logics. Cambridge, U.K.: Cambridge University Press, 1978.

Jacquette, Dale, ed. Philosophy of Mathematics: An Anthology. Oxford and Malden, Mass.: Blackwell, 2002.

Kitcher, Philip. The Nature of Mathematical Knowledge. New York: Oxford University Press, 1983.

Kneale, William, and Martha Kneale. The Development of Logic. Oxford: Clarendon, 1962.

Krüger, Lorenz et al., eds. The Probabilistic Revolution. 2 vols. Cambridge, Mass.: MIT Press, 1987.

Priest, Graham. Beyond the Limits of Thought. Cambridge, U.K.: Cambridge University Press, 1995.

Shapiro, Stewart, ed. The Limits of Logic. Aldershot, U.K.: Dartmouth, 1996. Source book, mainly oriented around model theory and higher-order logics.

van Heijenoort, Jean, ed. From Frege to Gödel: A Source Book in Mathematical Logic, 18791931. Cambridge, Mass.: Harvard University Press, 1967.

I. Grattan-Guinness