1. First foundations of mathematics

1.1. Introduction to the foundations of mathematics

What is mathematics

Mathematics is the study of systems of elementary objects, whose only considered nature is to be exact, unambiguous (two objects are equal or different, related or not; an operation gives an exact result...). Mathematics as a whole can be seen as «the science of all possible worlds» of this kind (of exact objects).
Mathematical systems are conceived as «existing» independently of our usual world or any particular sensation, but their study requires some form of representation. Diverse ways can be used, that may be equivalent (giving the same results) but with diverse degrees of relevance (efficiency) that may depend on purposes. Ideas may first appear as more or less visual intuitions which may be expressed by drawing or animations, then their articulations may be expressed in words or formulas for careful checking, processing and communication. To be freed from the limits or biases of a specific form of representation, is a matter of developing other forms of representation, and exercise to translate concepts between them. The mathematical adventure is full of plays of conversions between forms of representation, which may initiate us to articulations between mathematical systems themselves.

Theories

Mathematics is split into diverse branches according to the kind of systems being considered. These frameworks of any mathematical work may either remain implicit (with fuzzy limits), or formally specified as theories. Each theory is the study of a supposedly fixed system that is its world of objects, called its model. But each model of a theory may be just one of its possible interpretations, among other equally legitimate models. For example, roughly speaking, all sheets of paper are systems of material points, models of the same theory of Euclidean plane geometry, but independent of each other.

The word «theory» may take different meanings between mathematical and non-mathematical uses (in ordinary language and other sciences). A first distinction is by nature (general kind of objects); the other distinction, by intent (realism vs. formalism) will be discussed later.

Non-mathematical theories describe roughly or qualitatively some systems or aspects of the world (fields of observation) which escape simple exact description. For example, usual descriptions of chemistry involve drastic approximations, recollecting from observations some seemingly arbitrary effects whose deduction from quantum physics is usually out of reach of direct calculations. The lack of clear distinction of objects and of their properties induces risks of mistakes when approaching them and trying to infer some properties from others, such as to infer some global properties of a system from likely, fuzzy properties of its parts.

Pure mathematical theories, only describing exact systems, can be protected from the risk to be «false», by use of properly rigorous methods (formal rules) designed to ensure the preservation of exact conformity of theories to their models.

In between both, applied mathematical theories, such as theories of physics are also mathematical theories but the mathematical systems they describe are meant as idealized (simplified) versions of aspects of given real-world systems while neglecting other aspects; depending on its accuracy, this idealization (reduction to mathematics) also allows for correct deductions within accepted margins of error.

Foundations and developments

Any mathematical theory, which describes its model(s), is made of a content and is itself described by a logical framework. The content of a theory is made of components which are pieces of description (concepts and information, described in 1.3). A theory starts with a choice of foundation made of a logical framework and an initial version of its content (hopefully rather small, or at least simply describable). The components of this initial version are qualified as primitive.
The study of the theory progresses by choosing some of its possible developments : new components resulting from its current content, and that can be added to it to form its next content. These different contents, having the same meaning (describing the essentially same models), play the role of different presentations of the same theory. Any other possible development (not yet chosen) can still be added later, as the part of the foundation that could generate it remains. Thus, the totality of possible developments of a theory, independent of the order chosen to process them, already forms a kind of «reality» that these developments explore.

To express the properties of its models, the content of a theory includes a list of statements, which are formulas meant as true when interpreted in any model. Primitive statements are called axioms. Further statements called theorems are added by development to the content, under the condition that they are proven (deduced) from previous ones : this ensures them to be true in all models, provided that previous ones were. Theorems can then be used in further developments in the same way as axioms. A theory is consistent if its theorems will never contradict each other. Inconsistent theories cannot have any model, as the same statement cannot be true and false on the same system. The Completeness Theorem (1.9, 4.6) will show that the range of all possible theorems precisely reflects the more interesting reality of the diversity of models, which indeed exist for any consistent theory.
Other kinds of developments (definitions and constructions) which add other components beyond statements, will be described in 1.5, 4.8 and 4.9.

There are possible hierarchies between theories, where some can play a foundational role for others. For instance, the foundations of several theories may have a common part forming a simpler theory, whose developments are applicable to all.
A fundamental work is to develop, from a simple initial basis, a convenient body of knowledge to serve as a more complete "foundation", endowed with efficient tools opening more direct ways to further interesting developments.

Platonism vs Formalism

Mathematics, or each theory, may be approached in two ways (as further discussed in 1.9):

Many philosophers of mathematics carry obsolete conceptions of such views as forming a multiplicity of opposite beliefs (candidate truths) on the real nature of mathematics. But after examination, just remain these two necessary and complementary views, with diverse shares of relevance depending on topics :

By its limited abilities, human thought cannot directly operate in a fully realistic way over infinite systems (or finite ones with unlimited size), but requires some kind of logic for extrapolation, roughly equivalent to formal reasonings developed from some foundations ; this work of formalization can prevent possible errors of intuition. Moreover, mathematical objects cannot form any completed totality, but only a forever temporary, expanding realm, whose precise form is an appearance relative to a choice of formalization.

But beyond its inconvenience for expressing proofs, a purely formalistic view cannot hold either because the clarity and self-sufficiency of any possible foundation (starting position with formal development rules), remain relative: any starting point had to be chosen somehow arbitrarily, taken from and motivated by a larger perspective over mathematical realities; it must be defined in some intuitive, presumably meaningful way, implicitly admitting its own foundation, since any try to specify the latter would lead to a path of endless regression, whose realistic preexistence would need to be admitted.

The cycle of foundations

Despite the simplicity of nature of mathematical objects, the general foundation of all mathematics turns out to be quite complex (though not as bad as a physics theory of everything). Indeed, it is itself a mathematical study, thus a branch of mathematics, called mathematical logic. Like any other branch, it is full of definitions and theorems about systems of objects. But as its object is the general form of theories and systems they may describe, it provides the general framework of all branches of mathematics... including itself. And to provide the framework or foundation of each considered foundation (unlike ordinary mathematical works that go forward from an assumed foundation), it does not look like a precise starting point, but a sort of wide cycle composed of easier and harder steps. Still this cycle of foundations truly plays a foundational role for mathematics, providing rigorous frameworks and many useful concepts to diverse branches of mathematics (tools, inspirations and answers to diverse philosophical questions).

(This is similar to dictionaries defining each word by other words, or to another science of finite systems: computer programming. Indeed computers can be simply used, knowing what you do but not why it works; their working is based on software that was written in some language, then compiled by other software, and on the hardware and processor whose design and production were computer assisted. And this is much better than at the birth of this field.)

It is dominated by two theories:

Each one is the natural framework to formalize the other: each set theory is formalized as a theory described by model theory; the latter better comes as a development from set theory (defining theories and systems as complex objects) than directly as a theory. Both connections must be considered separately: both roles of set theory, as a basis and an object of study for model theory, must be distinguished. But these formalizations will take a long work to complete.

Set theory and Foundations of mathematics
1. First foundations of mathematics
1.1. Introduction to the foundations of mathematics
1.2. Variables, sets, functions and operations
1.3. Form of theories: notions, objects, meta-objects
1.4. Structures of mathematical systems
1.5. Expressions and definable structures
1.6. Logical connectives
1.7. Classes in set theory
1.8. Binders in set theory
1.9. Axioms and proofs
1.10. Quantifiers
Time in model theory
Set theory as unified framework
2. Set theory (continued) - 3. Algebra - 4. Arithmetic 5. Second-order foundations
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FR : Introduction au fondement des mathématiques