# Overview:

see also: What is Foundations of Mathematics? by Steve Simpson
 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.

# Branch Foundations:

 Just a mention of the few examples that come to my mind. Real mathematicians could doubtless do much better.
 Fundamental Concepts A branch of mathematics is likely to have one or more fundamental concepts which are definitive of the scope of the field. e.g. the concept of a group is fundamental to group theory. Probably quite a few other concepts would also be counted as fundamental to group theory, e.g. that of homomorphism.
 FUNDAMENTAL THEOREMS Many branches of mathematics also have a "fundamental theorem". fundamental theorem of arithmetic States that every number can be factorised into prime numbers in just one way. fundamental theorem of the calculus States that if you differentiate an integral then you get back the original function. fundamental theorem of algebra States that every polynomial equation has at least one root.

# Fundamental Concepts:

 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).
 Set Theory Set theory provides a conceptual framework which has been fundamental to pretty much the whole of mathematics during this century. The definition of the concept of function via sets and relations is the centerpiece of this conceptual framework. This was controversial while evolving in the 19th Century, and remains a point where changes are sometimes advocated which would have fundamental impact throughout mathematics.
 Category Theory Mathematics could be called the science of abstraction. During this century there have been advances in abstract algebra, leading first to universal algebra and then to category theory. These developments provide better ways of structuring mathematics, potentially transforming the look-and-feel of mathematics without affecting the substance of the underlying subject matter. Fundamental set theoretic concepts are superseeded by more structured and abstract versions, e.g. talk of functions may be supplanted by talk about morphisms, and the identification of functions with their graphs downplayed.
 Constructive Mathematics In constructive alternatives to classical mathematics the concept of function is radically changed resulting in material differences in the various parts of mathematics which depend upon it. Functions are no longer conceived as arbitrary graphs, but as certain kinds of rule. This weeds out of the mathematical population many exotic creatures, but also may introduce new and useful functions which are omitted from the classical firmament because not well-founded. It is moot whether this kind of fundamental conceptual change can be viewed as a change of subject matter, or whether yet more fundamntal issues are at stake.

# 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.
 Canons of Reasoning Mathematics is a deductive rather than an empirical science. Mathematicians will generally accept only those results for which they have (or could obtain) demonstrative proof. Codifying the rules of proof is therefore an important part of laying the foundations for mathematics. If you change the rules of reasoning you change the mathematics you can derive. Intuitionists do this deliberately, and end up with constructive rather than "classical" mathematics.
 Matters of Ontology It is essential to have some abstract objects about if you are to do mathematics. These can be implicit in the logical system, or provided through non-logical axioms. As well as, and prior to, providing basic concepts such as that of a function, set theory provides a complex universe of sets (many of which are then used as representatives of functions). Changes to the ontology are liable to affect the mathematics which is subsequently derived. In set theory, logic and ontology appear to be separated, but in other systems this isn't so clear. Intuitionists connect ontological excess with underlying logical principles.
 Absoluteness and Universality Logic is often said to be general and topic neutral. The first attempts, by Frege and Russell, to provide logical foundations to mathematics were based on the idea that the whole of mathematics could be developed from a neutral logic by defining mathematical concepts and then deriving mathematical theories. Since then many doubts have been cast upon this vision. There remain philosophical, technical and pragmatic reasons for the belief that underlying the syntactic babel there can be located an absolute notion of logical truth and absolute foundation systems which can provide a basis for the development of any mathematical subject domain.

© created 1998/7/24 modified 1998/7/31