Mathematics is to deductive reasoning what Science is to inductive reasoning. To have a good understanding of Mathematics we should have a good understanding of the difference between induction and deduction.

The kind of induction that we are talking about is mathematical induction. We might have called it scientific induction, except that our understanding of Science developed after mathematical induction. As we use induction in science, it becomes implicit in the things in the things we are talking about. When we stand in line and hear someone say, “next”, we know that ‘next’ will eventually refer to us. We don’t often stop to think that we are using an inductive concept.

The words that we use in scientific theories make use of induction (see Platonic forms, abstraction). If there is disagreement on what those words refer to, then scientists can modify the experiment to address those concerns (see peer review). In fact, science, occasionally, needs to measure something that has never been measured before, so applied science blends with engineering.

The analogous situation for Mathematics may shock mathematicians. Mathematics, eventually, needs to model things that have not been modeled before. Mathematics needs to use rules that have not been used before. Applied mathematics blends into philosophy. We may like Mathematics because of the certainty that deductive reasoning provides, but that certainty does not extend to the choice of rules.

Inductive concepts can be added to deductive systems as closures of existing axioms. Such axioms fix a definition of infinity. Such axioms cannot combine to form other forms of infinity. To make more stuff deductively true, we need to add axioms.

As an application of these ideas, consider that the arguments for ‘the completeness of the Real numbers’ are informal and self referential. Presaging modern mathematics was Cantor’s Cardinality. As Cardinality is one (not alone / only) method of extending the notion of magnitude / size to infinity, it is contingent and invalid. Since Cantor, we have discovered that the Continuum hypothesis cannot be proven true or false from ZFC. A course in analysis of single Real variable indoctrinates students to think of inductive concepts in ways they cannot work. The belief in a single set of axioms to rule them all is misguided.

My point of view might be called anti-logicism. As long as people have things to learn (from / with deductive reasoning), then we will want to know that the rules they are using are consistent. By ‘the incompleteness thesis’, we cannot know if a system is consistent within itself. Deductive truth depends solely on the rules. There is no deductive system that can uncover every deductive truth. We cannot escape the subtle (non-deductive) practice of evaluating alternative rules.

Maybe someone more talented will understand and explain better. Till then, I did my best.

The most important lesson of math: Multiple sets of rules are needed to do math. Deductive truth is relative to the rules chosen.

Mathematics is to deductive reasoning what science is to inductive reasoning. Notice both, how well deduction and induction work together and that infinity is an inductive concept. The kind of inductive concepts that we can formalize as axioms are closures of the other axioms. Such axioms are guaranteed to be compatible. A trade-off of such axioms is that, in practice, they make it harder to separate validity from contingency (necessity from possibility, when are definitions are warranted).

Here are some examples of such misunderstandings:

First, the Incompleteness theorems are not theorems. Incompleteness is so important that it is universally part of an undergraduate math curriculum. For it to be a theorem we would need a deductive system that formalizes all deductive systems. Such a system would be inconsistent.

Second, we cannot have a simple axiom to formalize the unorderedness of sets. The apparent simplicity of sets is deductively deceptive. It takes some effort to formalize order (and a lack of order). That is not the way AC is introduced to students.

Third, computation cannot be extended to formalize infinite processes. The theory of computation enables all combinations and permutations up to some length. To extend the theory of computation to infinity would lead to a deductive explosion (i.e. a contradiction). Alternatively, formalizing infinite procedures means that string encodings are no longer bounded. Unbounded string encodings need a formalism for infinite procedures, but infinite procedures need string encodings to be fixed / formalized. A self referential / logical loop, resulting in a contradiction.

Fourth, limit points do not validate the continuum. The continuum hypothesis is independent of ZFC, but CH was not intended as an axiom.

We cannot assume that a definition forms a set (see proper classes) or that a set of definitions can be used together (ZFC |-> limits versus NF |-> randomness ??). We have to let go of the dogma that a single set of axioms captures all mathematical concepts. Mathematicians calculate, prove, and model, but modeling is not a deductive process. A system cannot prove its own consistency, so we make it a subsystem by adding an axiom. So long as we use deductive reasoning to explore ideas, we will want to know that the rules we choose are consistent.

Let us recognize the asymmetry of deduction. All deductions, all proofs, and all computations each are semi-decidable inductive concepts. Every proof, deduction, and computation can be verified, but the lack there of cannot. Inductive concepts (as closures of deductive concepts) are computationally approximatizable (the computational complement of computational enumeration).

You might want more. I have grown old trying to reach out. I have yet to find anyone to talk to. I might not be able to fit in or give a satisfying explanation, but I can offer consolation. I offer a few suggestions.

A lot of effort goes into learning foundations (Set theory, First order logic, theory of computation). We should emphasize that definitions are usually contingent, so that we do not undermine formal methods.

We can introduce Topology as an abstraction of distance, which merges with the concept of connectedness. We can introduce to definition of Topological spaces alongside graphs for contrast. Pointless Topology can be introduced early.

I wonder if we could start abstract algebra with permutation groups. Emphasize group membership versus group action. Work upto Cayley’s theorem. Then the nuances of operations might be clearer and we would have an opportunity to introduce Universal algebra early.

Combinatorics may have many applications, but students might benefit to consider it in relation to computation / foundations. Where would Algebra be without groups and Analysis without nets?

We might discuss generalizing the notion of primeality to a minimal topological cover useing algebraic cycles. I thought that was interesting, but have never figured out where number therory fits into a four year curriculum.