Paradoxes and strange functions pushed nineteenth-century mathematicians to put logic under the microscope
Karl Weierstrass built continuous functions that have no slope anywhere, and flaws turned up in Euclid's supposedly airtight geometry. Unsettled, nineteenth-century mathematicians began turning logic into mathematics itself, writing axioms for arithmetic and geometry and asking whether the whole edifice could be proved consistent. The answers reshaped the field.
Mathematical logic studies formal logical systems with mathematical tools, asking how much they can express or prove, and uses logic to pin down sound reasoning and the foundations of mathematics. A 1977 handbook split it roughly into set theory, model theory, recursion or computability theory, and proof theory together with constructive mathematics. The boundaries blur: the technique called forcing crosses several areas, and category theory, which Saunders Mac Lane proposed as an alternative foundation, is usually kept separate.
Logic has roots in ancient China, India, Greece, Rome and the Islamic world. Aristotle's term logic dominated Western thought for millennia, and the Stoic Chrysippus began propositional logic. Leibniz and Lambert tried algebraic treatments in the 18th century with little influence. In the mid-19th century George Boole and Augustus De Morgan gave logic a systematic mathematical form, Vatroslav Bertić independently algebraised it in 1847, and Charles Sanders Peirce added relations and quantifiers in papers from 1870 to 1885. Gottlob Frege's 1879 Begriffsschrift, a turning point, stayed obscure until Bertrand Russell promoted it, and Ernst Schröder's three volumes of 1890 to 1905 summed up the century.
Foundations were the driving worry. Giuseppe Peano wrote axioms for the natural numbers without knowing Frege's work, while Richard Dedekind characterised them by their induction properties and in 1858 defined real numbers through cuts of rationals. Nikolai Lobachevsky had shown in 1826 that the parallel postulate was independent, and David Hilbert later gave geometry a complete axiom set building on Moritz Pasch. Georg Cantor proved the reals outnumber the naturals, and in 1891 introduced his diagonal argument to show no set matches the size of its power set.
Paradoxes in informal set theory raised fears that mathematics might be inconsistent. Hilbert's 1900 list of 23 problems opened with the continuum hypothesis and the consistency of arithmetic, and his 1928 decision problem asked for a mechanical test of provability. Kurt Gödel, Gerhard Gentzen and others answered partly, and modern work often asks which parts of mathematics particular systems can formalise.
Source: Mathematical logic
