The study of logic underwent a profound rebirth at the turn of the twentieth century. While logicians were traditionally concerned with the correctness of reasoning in ordinary language, Frege introduced formal languages as the primary medium and objects of logical enquiry. The shift from ordinary language to artificial languages containing formal expressions elevated the study of mathematics and philosophy to a new level of rigour and marked the beginning of the computer age.

But what exactly are formal expressions? Although the ontological nature of mathematical objects, such as numbers, has been a central topic in philosophy since antiquity, a systematic study of the ontology of formal expressions is hitherto lacking in the literature.

The aim of this talk is threefold. First, I will show that within the last century, the study of formal languages underwent a "structuralist turn". I will then argue that recent attempts to make the structuralist view of formal languages explicit are flawed. Finally, I will show how the structuralist view can be stated correctly.

The radical development of formal methods in logic in the past two centuries has brought logic closer to mathematics, and, at the same time, accentuated the tension between the previously disparate fields. The driving foundationalist agenda, which attained dominance at the turn of the 20th century, is at tension with ordinary mathematical practice - as it threatens its autonomy by stripping it of ultimate authority as to what counts as valid proof.

In a recent book, Juliette Kennedy draws attention to this tension. The notion of formalism freeness has a central role in Kennedy's work, which, put concisely, means “the suppression of any […] aspects of a logic or formalism, except semantics”. In my talk I will pause on the distinction between formal and natural language that I take to be at the heart of the matter. Kennedy’s notion of formalism freeness that pertains to the subject matter of mathematical practice, but, arguably, it is first of all the freedom of the mathematician that it is at stake - the freedom to use their natural language. I then propose a reverse perspective by which formal language is where the freedom of the mathematician is most starkly exercised.

Tarski notoriously argued, from consideration of the Liar paradox, that no consistent language can be semantically closed, and our only option for a consistent theory of truth is a hierarchy of languages. These claims have been subject to much criticism over the years. Many hold them to be prima facie implausible, and some hold that Tarski’s notion of a consistent language is itself a confusion. I, on the other hand, have long advocated for a highly revised form of a Tarskian hierarchy. In this paper, I return to the question of how to formulate claims about semantic closure, and I argue that a properly formulated hierarchical theory of truth is more plausible and natural than it is usually taken to be.