# 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(x1,..., xn)), true for a unique value of y for any chosen values of x1,..., xn). 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:

 Set theory and Foundations of mathematics (general table of contents) 1. First foundations of mathematics (detailed table of contents) Philosophical aspects (Each subsection roughly uses those left and up) 1.1. Introduction to the foundations of mathematics 1.2. Variables, sets, functions and operations Intuitive representation and abstraction Platonism vs Formalism 1.3. Form of theories: notions, objects, meta-objects Realistic vs. axiomatic theories 1.4. Structures of mathematical systems 1.5. Expressions and definable structures ⇨Time in Model theory The time of interpretation The metaphor of the usual time The finite time between expressions 1.6. Logical connectives 1.7. Classes in set theory The infinite time between theories Zeno's Paradox 1.8. Binders in set theory Time in Set theory Expansion of the set theoretical universe Can a set contain itself ? 1.9. Quantifiers The relative sense of open quantifiers Interpretation of classes Classes in an expanding universe Concrete examples 1.10. Formalization of set theory 1.11. Set generation principle Justifying the set generation principle Concepts of truth in mathematics Alternative logical frameworks ⇨ 2. Set theory (continued)
Other languages:
FR : Aspects philosophiques