# The Foundations of Mathematics

## Overview:

 Mathematics is a logical science, cleanly structured, and well-founded. Here we look at those foundations.
 What is a "foundation" for mathematics? We discuss some of the ways the word "foundation" is used in relation to mathematics. History of Foundations An assortment of historical cameos, starting in ancient Greece, running through the turbulent present into an exiting future. Some FOM postings This link searches FOM for my contributions.
 Logical Foundation Systems The methods of mathematics are deductive, and logic therefore has a fundamental role in the development of mathematics. Suitable logical frameworks in which mathematics can be conducted can therefore be called logical foundation systems for mathematics. Alternative Foundation Systems A bewildering variety of alternative foundation systems are superficially surveyed.

## What is a "foundation" for mathematics?:

 We discuss some of the ways the word "foundation" is used in relation to mathematics.
 Branch Foundations Each branch or field of mathematics may have its own foundational elements, special to the field. These may be the fundamental concepts investigated in the field, fundamental results on which most other results are based, or pervasive methods.
 Fundamental Concepts There are some mathematical concepts which pervade not just one branch of mathematics but the whole of mathematics. An obvious example is the concept of a function or that of a homomorphism. Variation in these concepts may either provide different ways of doing mathematics (e.g. category theory), or may lead to different kinds of mathematics (e.g. constructive mathematics).
 Logical Foundations The methods of mathematics are deductive, and logic therefore has a fundamental role in the development of mathematics. Suitable logical frameworks in which mathematics can be conducted can therefore be called logical foundation systems for mathematics.

## Logical Foundations for Mathematics:

 The methods of mathematics are deductive, and logic therefore has a fundamental role in the development of mathematics. Suitable logical frameworks in which mathematics can be conducted can therefore be called logical foundation systems for mathematics.
 Paradigms There are many alternative logical foundations for mathematics. They differ in some cases just in detail or strength. In other cases the differences are more fundamental, possibly representing radically divergent views on the nature of mathematics. Dimensions We consider five different characteristics or dimensions of logical foundation systems, sometimes clearly separated, other times not so. The formal the semantic, the logical, the ontological and the conceptual.
 Formal The formal aspect concerns the mathematical theorems which can be proven using the foundation system. Foundation systems are partially ordered according to their proof-theoretic strength. Semantic Gödel's first incompleteness theorem guarantees that truth and theoremhood do not coincide in any foundation system adequate for mathematics. It is therefore desirable to have an account of the intended meaning of the language independent of the definition of formal derivability.
 Logical In some cases it is possible to separate out a part of the system which is concerned with logic and independent of matters ontological. Ontological Ontology is an important part of semantics, and differences both paradigmatic and in detail, are likely to be reflected in or caused by ontology. Conceptual Over and above the logical and ontological features which determine the strenght of the system there is likely to be some conceptual apparatus which provides the first stages in developing mathematics.

## History of the Foundations of Mathematics:

 An assortment of historical cameos, starting in ancient Greece, running through the turbulent present into an exiting future.
 Greece During the period from about 600 B.C. to 300 B.C. , known as the classical period of Greek mathematics, mathematics was transformed from an ecclectic collection of practical techniques into a coherent structure of deductive knowledge. Axiomatic Method Invented by the ancient Greeks and used (on and off) ever since, with one or two tweaks to take account of modern logic. History of Rigour A two thousand year sketch of how rigour in mathematics has evolved to the present day.
 The Formalisation of Mathematics During the period from about 1821 to 1908, the somewhat doubtful foundations for analysis were sorted out and the first formal logical foundation systems for mathematics were devised. Formality and Rigour in 20th Century Mathematics A personal view about a couple of interesting features of modern mathematics. 20th Century Schools Logicism, formalism, intuitionism, and the sequel.
 History of Logical Foundations Pictorial overview, focussing on the logical side. Category Theory and Foundations Does category theory offer a superior modern alternative to old fashioned set theory? Are Foundations Necessary? Isn't it just so much better to have lots of languages for different domains, each with their own logic, rather than shoe horn everything into a foundation system?

## Alternative Foundation Systems:

 A bewildering variety of alternative foundation systems are superficially surveyed.
 What's wrong with ZFC? A discussion of reasons for considering alternatives to ZFC as a foundation for mathematics.