It is a well-known, but uneasy, admission that the foundations of mathematics seem to be inherently circular. Indeed, we may for example list out the axioms for the positive integers, say using the Peano formulation, but in writing down this list we already use numbers: identifying the first axiom, then the second axiom, and so on. Even if we argue that we need only formally specify a set of axioms rather than a list, this still makes the axiomatic formulation of sets circular, whether with Zermelo-Fraenkel or another system.
As another example, the organization of different “logics” in mathematical logic as zeroth-order, first-order, higher-order, etc., inherently involves the positive integers. Yet the positive integers are only formulated within these logics.
How might we address these conundrums?
In this post, I’ll discuss my thoughts on the philosophical foundations of this topic, and I’ll also consider and address some related questions.
The Informality of Formal Language
My first contention is one that may sound ironic: formal language cannot be completely formal.
I will establish my argument using contradiction. Say we had such a completely formal description of formal language. Content written in such a language would consist of a sequence of symbols, which are elements of a fixed set which we designate to be the “alphabet.” Yet the terms “set” and “sequence” themselves must be defined purely in terms of formal language. This would be circular, which means that any proof involving these ideas could not terminate. Since we expect proofs to be read by humans, and therefore expressed in a finite amount of space, this would produce a contradiction.
Since formal language cannot be completely formal, the understanding of what language is formal and what is not, must be informal, for otherwise such a formal understanding would itself allow a completely formal definition of formal language. In fact, my judgments above that proof by contradiction “works” for formal language, and that proofs need to be written down in a finite amount of space, were inherently informal.
Thus, formal language can only “work” on top of a foundation of informal language.
However, let us be clear: informal language is not any less “real.” In fact, informal language is how we often describe everyday, non-abstract occurrences, which are of course very “real.” This informal language is what can then give some “real” meaning to the definition of formal language.
To go further, I argue that notions such as grouping (sets), counting (positive integers), and reasoning (logic), must be fundamentally informal. In fact, other notions must be understood informally as well. For example, the Peano axioms for the positive integers use predicates that are dependent on arguments, and these use function syntax, even before functions are introduced with domain and range the set of positive integers. Thus, functions must be understood informally, and similarly so must relations.
Formal Models of Informal Notions
It is a favorite pastime of mathematicians to construct formal models, whether with definitions or sets of axioms (or both), that capture some “intuitive” aspect, notion, or idea which they informally understand.
There is no reason this can’t be done for notions which we argue must be first informally understood. For concepts such as sets and positive integers, that is exactly what formulations such as Peano and Zermelo-Fraenkel do.
However, it is important to note the ordering of activities here. These notions must be understood informally first, and only later can we go back and construct formal models which capture some aspect of our intuitive understanding of these notions.
And we, humans, are still always the judges of whether our formal models conform to our intuitive understandings of the concepts they model. Furthermore, such judgments must be made informally, outside of formal logic.
