Philosophical aspects of the foundations of
mathematics
To complete our initiation
to the foundations of mathematics, the following pages of
philosophical complements (from this one to Concepts of truth in
mathematics), will present an overview on some of the deepest
features of the foundations of mathematics: their philosophical and
intuitive aspects (much of which may be implicitly understood but
not well explained by specialists, as such philosophical issues are
not easily seen as proper objects of scientific works). This
includes
- How, while independent of our time, the universe of
mathematics is still subject to a flow of its own time ;
- The deep meaning of the difference between sets and classes,
in relation to that time; thus, the deep reason for the use of bounded
quantifiers in set theory.
- The justification of the set
generation principle
These things are not necessary for Part 2 (Set Theory, continued)
except to explain the deep meaning and consequences of the fact that
the set exponentiation or power set (2.6) is not justifiable by the
set generation principle. But they will be developed and justified
in more details in Part 4 (Model Theory).
Intuitive representation and abstraction
Though mathematical systems «exist» independently of any particular
sensation, we need to represent them in some way (in words, formulas
or drawings). 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 usually first appear
as more or less visual intuitions, then are expressed as formulas
and literal sentences for careful checking, processing and
communication. To be freed from the limits of a specific form of
representation, the way is to develop other forms of representation,
and exercise to translate concepts between them. The mathematical
adventure itself is full of plays of conversions between forms of
representation.
Platonism vs Formalism
In this diversity of approaches to mathematics (or each theory), two
philosophical views are traditionally distinguished.
- The Platonic view (also called idealistic)
focuses on the worlds or systems to study, seen as preexisting
mathematical realities to be explored (or remembered, according
to Plato). This is the approach of intuition that smells the
global order of things before formalizing them.
- The formalistic view focuses on language, rigor
(syntactic rules) and dynamical aspects of a theory, starting
from its foundation (formal expression), and following the rules
of development.
Philosophers usually present them as opposite, incompatible belief
systems, candidate truths on the real nature of mathematics.
However, both views instead turn out to be necessary and
complementary aspects of math foundations. Let us explain how.
Of course, human thought having no infinite abilities, cannot
fully operate in any realistic way, but only in a way roughly
equivalent to formal reasonings developed from some foundations ;
this work of formalization can prevent the possible errors of
intuition.
But a purely formalistic view cannot hold either because
- The clarity and self-sufficiency of any possible foundation
(any starting position with formal development rules), remain
relative: any starting point had to be chosen more or less
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 have to be admitted.
- Most of the time, works are only partially formalized: we use
semi-formal proofs, with just enough rigor to give the feeling
that a full formalization is possible, yet not fully written; an
intuitive vision of a problem may seem clearer than a formal
argument.
Another reason for their reconciliation, is that they are not in any
global dispute to describe the whole of mathematics, but their
shares of relevance depends on specific theories under
consideration.
The form of mathematical theories
The main useful logical frameworks
for mathematical theories,
from the weakest (less expressive) to the strongest (most
expressive), except set theory, will be:
- Algebraic theories;
- First-order logic
- Duality theories (that is a fuzzy range of theories with some
common features);
- Second-order logic;
- Higher-order logic;
We described the general form of mathematical theories in sections
1.1,
1.3,
1.4,
1.5,
1.6,
1.8,
1.9 for the two
most important frameworks : first-order logic, and the framework of set
theory (which is specific, with definiteness classes, and binders depending
on sets). The relation between both escapes the above hierarchical
order as we have irreversible correspondences between them in both
ways (not the inverse of each other):
- A natural inclusion of all first-order theories into set
theory, viewing their possible models as objects in a common
model (universe) of set theory (1.3, 1.4). This is the usual
view of ordinary mathematics, studying many systems as "sets
with relations or operations such that...", with possible
connections between them.
- Another procedure (sections 1.9, 1.10, with a more restricted
interest to specialists of mathematical logic) converts set
theory into first-order logic.
Let us sum up our description: once chosen a formalism, a theory is
specified by its content (vocabulary and axioms), as follows. Every
theory is made of 3 kinds of components. Components in the latter
2 kinds are finite systems made from those of the previous kind :
- A list of types, whose translations into set theory
are names of sets. For example, geometry can be seen
with 2 types: «points» and «lines».
- A list of structure symbols: these aim to designate structures,
relating object with specific types to give their roles to the objects of each
type with respect to those of the same or other types. A structure may be
- A relation, which may be seen as an operation with value «true»
or «false» (the relation symbol comes with the list of arguments and the type of each)
- An operation between variables, with values of also one specified type
(an n-ary operation f may be reduced to the (n+1)-ary relation
(y=f(x_{1},..., x_{n})), true
for a unique value of y for any chosen values of
x_{1},..., x_{n}).
A constant
is a nullary operation (it has no argument).
- A technique to make more powerful structures will be explained with second-order logic.
- A list of axioms. Each axiom is a ground formula : a
system of occurrences of symbols among structure symbols,
equality (=), logical connectives (1.6) and quantifiers (∀and ∃,
1.9); these formulas are supposed to be true when these symbols
are interpreted in a given system.
But for having any interest for most practical purposes, a
theory should be such that we cannot build from this, any
of the 4th kind of components:
- A contradiction of a theory, is a system of formulas
based on its axioms, forming a proof of the formula 0 (False), as explained in 1.6.
Realistic vs. axiomatic theories in mathematics and other
sciences
Interpretations of the word «theory» may vary between mathematical
and non-mathematical uses (in ordinary language and other sciences),
in two ways.
Theories may differ by their object and nature:
- Pure mathematical theories, are mathematical theories
considered for the pure sake of mathematics, without any
non-mathematical intentions.
On the contrary, theories outside
mathematics, try to describe some real systems (fields of
observation, parts of the outside world, that are not purely
mathematical systems). They may be of 2 sorts:
- Applied mathematical theories are also mathematical
theories (i.e. expressed in rigorous ways) but the mathematical
systems they describe are conceived as idealizations of aspects
of given real-world systems (neglecting other aspects); insofar
as it is accurate, this idealization (reduction to mathematics)
also allows for correct deductions within accepted margins of
error.
- Non-mathematical theories describe qualitative
(non-mathematical) aspects of the world. 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.
Theories may also differ by whether Platonism or formalism best
describes their intended meaning :
A realistic theory aims to
describe a fixed system given from an independent reality,
so that any of its ground formulas (statements) will be either
definitely true or definitely false as determined by this system
(but the truth of a non-mathematical statement may be ambiguous,
i.e. ill-defined for the given real system). From this intention,
the theory will be built by providing an initial list of
formulas called axioms : that is a hopefully true
description of the intended system as currently known or guessed.
Thus, the theory will be true if all its axioms are indeed true on
the intended system. In this case, its logical consequences
(theorems, deduced from axioms) will also be true on the intended
model.
This is usually well ensured in pure maths, but may be speculative
in other fields. In realistic theories outside pure mathematics,
the intended reality is usually a contingent one among alternative
possibilities, that (in applied mathematics) are equally possible
from a purely mathematical viewpoint. If a theory (axioms list)
does not fit a specific reality that pure mathematics cannot
suffice to identify, this can be hopefully discovered by comparing
its predictions (logical consequences) with observations : the
theory is called falsifiable.
An axiomatic theory is a theory given with an axioms
list that means to define the selection of its models
(systems it describes), as the class of all systems where these
axioms are true. This may keep an unlimited diversity of models,
that remain equally real and legitimate interpretations. By this
definition of what «model» means, the truth of the axioms of the
theory is automatic on each model (it holds by definition and is
thus not questionable). All theorems (deduced from axioms) are
also true in each model.
In pure mathematics, the usual features of both possible roles of
theories (realistic and axiomatic) automatically protect them from
the risk to be «false» as long as the formal rules are respected.
Non-realistic theories outside pure mathematics (where the
requirement of truth of theorems is not always strict, so that the
concept of axiom loses precision) may either describe classes of
real systems, or be works of fiction describing imaginary or
possible future systems. But this distinction between real and
imaginary systems does not exist in pure mathematics, where all
possible systems are equally real. Thus, axiomatic theories of pure
mathematics aim to describe a mathematical reality that is existing
(if the theory is consistent) but generally not unique.
Different models may be non-equivalent, in the sense that
undecidable formulas may be true or false depending on the model.
Different consistent theories may «disagree» without conflict, by
being all true descriptions of different systems, that may equally
«exist» in a mathematical sense without any issue of «where they
are».
For example Euclidean geometry, in its role of first physical
theory, is but one in a landscape of diverse geometries that are
equally legitimate for mathematics, and the real physical space is
more accurately described by the non-Euclidean geometry of General
Relativity. Similarly, biology is relative to a huge number of
random choices silently accumulated by Nature on Earth during
billions of years.
Realistic and axiomatic theories both appear in pure mathematics, in
different parts of the foundations of mathematics, as will be
presented in the section on the truth concepts in
mathematics.
But let us first explain the presence of a purely mathematical flow
of time (independent of our time) in model theory and set theory.
Once 1.5 is read, continue with Time
in model theory:
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