Assume we are developing the curriculum for an extracurricular math program for an advanced student. How could we introduce formal reasoning and the axiomatic method, two cornerstones of the practice of math today, as early as possible? And specifically, what are the implications for this from the philosophical discussion of the formal foundations of mathematics, in my post On the Circularity of Mathematics?
One of the points that I discussed there is that some notions must be first taken informally, before we construct formal models of them. This would need to be done with some of the basic concepts of math. As far as I can currently tell, this includes such concepts as the positive integers, sets, functions, and relations.
However, we would want to quickly get to a point where we are doing logic that is more or less formal, and I believe we can do that quickly once we have a small informal foundation of these concepts.
In implementing this, one point to keep in mind is the following. When we learn science in school, we learn that the scientific method is the fundamental process. We don’t expect that students perform the types of experiments that physicists commonly perform today – in fact, often times they wouldn’t even have the equipment necessary to do so. But we still perform small experiments in class that give students the understanding and flavor of the approach used in the field.
In a similar way, there is a priori no reason that we have to use the final formulations of the Peano or Zermelo-Fraenkel axioms in introducing the axiomatic method to beginning students. Mathematicians typically like to construct axioms that are as basic or “primitive” as possible, but this can hinder progress for students first using this approach, so instead we can use less primitive axioms in educational settings.
As a concrete example, if we were teaching arithmetic (in the computational, not number theory, sense) or algebra, we can use the axiomatic method and assume some properties of operations (such as commutativity) as axioms, as opposed to starting with Peano’s successor function.
With this, we can introduce formal logic earlier on, which I believe is a worthy goal, especially for advanced students, because it is a subject that is fundamental for practicing mathematicians. The axiomatic method is as fundamental to math as the scientific method is to science, and we should make students more aware of that.
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EDIT (2025): This post was written a long time ago; now, I would view these ideas from a slightly different angle.
I would argue that formal logic and “informal” proofs of properties like commutativity are both important, and not just for advanced students but for everyone.
Really, the crux of the issue is that math isn’t taught rigorously early on — rigor here including both formal and informal logic (see my philosophical post on the definition of mathematical rigor for more on this) — and as a result lots of important points are glossed over, resulting in students being misled and/or confused. Instead, teachers should really focus more on clarifying why certain statements should actually be proven rigorously when the reason is not obvious, and focus less on proofs that always restart from basic first principles (which can be a common student complaint in subjects like geometry.) We shouldn’t hinder students with what to them could seem like pedantism or busy work, but we should also make sure that they understand the necessity of fully rigorous proofs.
