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.
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
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.
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.
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.
Platonism vs Formalism
Mathematics, or each theory, may be approached in two ways (as further discussed
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 :
- 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.
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:
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.
Introduction au fondement