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.

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.

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.

The study of the theory progresses by choosing some of its possible

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.

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.

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

- The
*Platonic*or*realistic*view, considers the mathematical realm or some particular described systems, as preexisting realities to be explored (or remembered, according to Plato). This is the approach of intuition which by imagining things, smells their order before formalizing them. - A
*formalistic*or*logicist*view focuses on language, rigor (syntactic rules) and dynamical aspects of a theory, starting from its formal foundation, and following the rules of development.

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.(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:-
**Set theory**describes the universe of «all mathematical objects», from the simplest to the most complex such as infinite systems (in a finite language). It can roughly be seen as one theory, but in details it will have an endless diversity of possible variants (indeed differing from each other). -
**Model theory**is the study of theories (their formalisms as systems of symbols), and systems (possible models of theories).**Proof theory**completes this by describing formal systems of rules of proofs. While these are usually meant as general topics (admitting variants of concepts), the combination of both can be specified into precise versions (mathematical theories) called*logical frameworks*, each giving a precise format of expression for a wide range of possible theories, and a format in which all proofs in any of these theories can in principle be expressed. There is an essentially unique preferable logical framework called*first-order logic*, by which the concepts of theory, theorem (as provable statement) and consistency of each theory, find their natural mathematical definitions; but other logical frameworks are sometimes needed too.

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 |
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1. First
foundations of mathematics
⇨ 1.2. Variables, sets, functions and
operations1.1. Introduction to the foundations of mathematics |
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2. Set
theory (continued) - 3. Algebra - 4. Arithmetic |
5. Second-order foundations |

FR : Introduction au fondement des mathématiques