A great deal of philosophical questioning splits into two disciplines: epistemology and ontology. Roughly speaking, epistemology addresses questions of knowledge: what considered valid knowledge? invalid knowledge? what is a good justification? bad justification? On the other hand, ontology addresses questions of existence or being: what is considered to exist? not exist? what constitutes the identity of an object? I use these two aspects, in philosophy, to frame my presentation of the foundations of mathematics. Namely, in contemporary mathematical practice, the primary mode of demonstrating a mathematical truth is to provide a proof of some proposition about mathematical objects. Hence, the determination of what constitutes a valid proof in mathematics fixes an epistemology and the determination of what constitutes a mathematical object fixes an ontology. These two determinations correspond to the two halves of the foundations of mathematics: first-order logic and Zeremelo-Fraenkel with Choice (ZFC) set theory.

First-order logic is the formal language in which all mathematical truths, may in principle, be stated. Namely, first-order logic only allows propostions formed from primitive terms, predicates, variables, and the words “and”,“not”,“or”,“implies”, “for all”, “\(=\) (equals)” and “there exists”. Further, first-order logic restricts reasoning to only deductive rules of inference such as modus ponens (i.e. A is true and A implies B is true, then B is true) and and-elimination (i.e. A and B is true, then B is true). A formal proof is a list of statements in first-order logic that are linked by these deductive rules of inference. That is, the valid forms of knowledge are dictated by statements that are expressible in first order logic. The deductive rules of inference that constitute a proof define the standard for good justification in mathematics. In this way, first-order logic through defining the language for valid statements and the rules for providing good justification can been seen as determining an epistemology for mathematics.

Zeremelo-Fraenkal with Choice (ZFC) set theory is a list of axioms stated in first-order logic with predicate \(\in\) denoting’is a member of’. Roughly speaking, a set is considered to be a collection of objects. However, as the paradoxes of the so-called ‘foundational crisis’ of the 20th century (e.g. Russel’s Paradox) demonstrate, this notion is not a definition that yields consistent results. For this reason, ZFC is used to formalize and lay down the rules for how the intuitive notion of a ‘set’ is supposed to operate in mathematics. Most objects that mathematicians talk about follow the properties stated by ZFC set theory. Each axiom in ZFC, with the exception of the axiom of extensionality, establishes the existence of some set. For example, the empty set axiom states that there exists a set with no members (i.e. \(\exists B \forall x (x \not\in B)\)) Another example is the power set axiom which states that for every set \(A\), all the subsets of the set form a set called the power set of \(A\). Further, the exceptional axiom, the axiom of extensionality provides a critrion for the identity of a set (i.e. when two sets are actually the same set). Since every axiom (except one) establishes the existence of some set and most mathematical objects follow these rules, ZFC set theory can been seen as fixing the ontology of mathematics.

The connection between first-order logic (the epistemology) and ZFC set theory (the ontology) is established on the metamathematical level by two theorems in mathematical logic called Soundnesss and Completeness. I will address this connection in my next post.