Foundations of mathematics

The foundations of mathematics are the logical frameworks that ensure mathematical theories remain consistent. Without these foundations, mathematics could generate self-contradictory ideas. These frameworks provide reliable concepts for theorems, proofs, and algorithms. They also allow for the philosophical study of how mathematical structures relate to reality. In modern mathematics, basic concepts like numbers, points, and lines are not defined as abstractions from the physical world. Instead, they are defined by their basic properties, which are known as axioms.

To understand how these foundations work, one must look at the process of mathematical assertion. An assertion is considered true only if it is a theorem. A theorem is a statement proven from true premises through a sequence of syllogisms. These syllogisms are formal inference rules. The premises used in these proofs must be either previously proven theorems or self-evident assertions called axioms or postulates. This systematic approach is known as the axiomatic method. While Aristotle believed axioms were true because they were self-evident or based on experiments, the modern axiomatic method focuses on the fact that the proof shows the axioms imply the theorem.

Mathematical foundations have evolved through several distinct stages. The first stage involved the early logic of ancient Greece. Philosophers like Aristotle laid down the rules for organizing knowledge using primitive concepts and definitions. This was later applied systematically by Euclid in his treatise, the Elements. The second stage arrived in the 17th century with the development of infinitesimal calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently created this field to study variable quantities and moving points. The third stage was a period of intense scrutiny during the 19th century. This era saw the rise of real analysis and the attempt to provide rigorous bases for calculus.

History shows that these foundations were not always stable. The Pythagorean school originally insisted that only natural numbers and their ratios existed. They were shocked to discover that the ratio of a square's diagonal to its side was not a ratio of two natural numbers. This led to the concept of irrational numbers. Later, the 17th-century introduction of calculus introduced new concepts like continuous functions, derivatives, and limits. These were based on infinitesimals, which are hypothetical numbers infinitely close to zero. The philosopher George Berkeley famously criticized these as "the ghosts of departed quantities" because they lacked formal definitions.

The lack of rigor in calculus eventually led to the foundational crisis of mathematics at the end of the 19th century. As mathematicians made progress, they encountered seemingly paradoxical results. These puzzles challenged the general confidence in the truth of mathematical results. To resolve this, a new discipline emerged: mathematical logic. This discipline includes set theory, model theory, proof theory, and computability theory. It also incorporates parts of modern computer science. The resolution of this crisis helped stabilize mathematics into a coherent and valid framework.

During the 19th century, specific efforts were made to define the building blocks of math more precisely. Augustin-Louis Cauchy worked to give rigorous bases to calculus by rejecting unproven algebraic properties. Karl Weierstrass later formalized the epsilon-delta definition of limits, which is still used today. He also discovered "pathological" functions that were continuous but nowhere-differentiable, which contradicted earlier intuitions. To define real numbers, Richard Dedekind developed the concept of Dedekind cuts in 1872. Around the same time, Georg Cantor provided a different definition using equivalence classes of Cauchy sequences.

Today, the foundations of mathematics are highly sophisticated and interconnected. The modern framework relies on the systematic use of the axiomatic method and set theory. Specifically, much of mathematics is based on Zermelo–Fraenkel set theory with the axiom of choice. Another important foundation is type theory, which is frequently used in computer proof assistants. While mathematical concepts are now defined by axioms rather than physical reality, physical reality still guides mathematicians. They use the real world to choose which axioms to use and to find interesting theorems to prove.