Philosophy of mathematics

Philosophy of mathematics is a specialized branch of philosophy. It examines the fundamental nature of mathematics and its connections to other fields. It focuses on areas like epistemology, which is the study of knowledge, and metaphysics, which explores the nature of reality. Central questions involve whether mathematical objects are purely abstract entities. Philosophers also ask if these objects are concrete in some way. They investigate how such objects relate to our physical reality. This field explores if math is a product of the human mind or a reality that exists independently.

Mathematical reasoning requires a high standard known as rigor. This means that definitions must be absolutely unambiguous. Proofs must be reducible to a succession of inference rules or syllogisms. These processes must function without using empirical evidence or intuition. The rules for rigorous reasoning were established by ancient Greek philosophers under the name of logic. While logic is not exclusive to mathematics, the standard of rigor is much higher in math. For many centuries, logic belonged to philosophy rather than being studied by mathematicians.

Around the end of the 19th century, mathematics faced a foundational crisis. Several paradoxes made the logical foundations of mathematics questionable. Some results contradicted common intuition. For example, non-Euclidean geometries showed that the parallel postulate could be wrong. The Weierstrass function is continuous but nowhere differentiable. Georg Cantor studied infinite sets and discovered different sizes of infinity, called infinite cardinals. Most strikingly, Russell's paradox showed that the phrase "the set of all sets" is self-contradictory. These issues challenged the validity of the whole mathematical system.

To solve these problems, mathematicians proposed different logical frameworks. One method is constructive mathematics, which requires an explicit example for every existence theorem. Another is intuitionistic logic, which excludes the law of excluded middle and double negation elimination. These logics use fewer inference rules than classical logic. Classical logic was originally a first-order logic. This meant quantifiers could not be applied to infinite sets. For instance, the sentence "every set of natural numbers has a least element" was nonsensical in that formalization. This led to the development of higher-order logics used today.

These foundational problems were eventually resolved through mathematical logic. A formal theory consists of a formal language and a set of basic assertions called axioms. It also uses inference rules to produce new assertions from known ones. A theorem is either an axiom or an assertion obtained via an inference rule. The Zermelo–Fraenkel set theory with the axiom of choice, known as ZFC, is a higher-order logic. Most mathematics is restated within ZFC. Other proposed foundations can also be modeled inside this framework. In this context, a proof is simply correct or erroneous.

Mathematics is also deeply connected to the physical sciences. Scientists use mathematical models to represent phenomena and make predictions. The accuracy of a prediction depends on the adequacy of the model rather than mathematical truth itself. For example, Einstein's general relativity replaced Newton's law of gravitation to explain the perihelion precession of Mercury. Mathematics is often considered falsifiable because a single counterexample can disprove a theory. Mathematicians also use experimentation, such as computation or studying representations of objects. The mathematician Gauss once described his process as "systematic experimentation."

Physicist Eugene Wigner identified a phenomenon called the unreasonable effectiveness of mathematics. This describes how pure mathematical theories often have applications in the physical world. These applications might involve phenomena unknown when the theory was first created. For example, prime factorization was discovered over 2,000 years before its use in the RSA cryptosystem for internet security. Similarly, the ancient Greek study of ellipses as conic sections predated Kepler's discovery of planetary trajectories by nearly 2,000 years. Even non-Euclidean geometry and manifolds, once seen as disconnected from reality, became essential to Einstein's theory of relativity.

Mathematical discoveries can even drive the direction of physics research. The equations of certain theories have produced unexplained solutions that suggest new particles. This occurred with the discoveries of the positron and the baryon. In both cases, specific experiments later confirmed these predicted particles. The history of mathematics is also marked by different schools of thought. In the 20th century, formalism, intuitionism, and logicism emerged to address concerns about certainty. These schools attempted to resolve the crisis of foundations or redefine the status of mathematical knowledge.