What inference mechanisms are used for justification in mathematics?

Question

I am aware of two major modes of reasoning used for justification of belief: deductive and inductive. Whereas physics relies on induction, mathematics seems to rely exclusively on deductive inference, which ensures the soundness of its conclusions. Is that correct? Are there other modes of inference used in mathematics?

Also, to be precise, what I am after here are the admissible modes of inference used for justification in the final product of mathematics, not the types of inferences mathematicians use in their daily work to generate mathematics.

For inductive reasoning, what I have in mind here is what is explaining in this entry of the SEP.

(if there is nothing in mathematics but deduction, mathematics is an extension of logic, and there is no other way to see the situation)

Answer 1

As a point of pedantry, mathematics doesn't use any mode of reasoning, mathematicians use modes of reasoning. And, naturally, like all people, they use a mixture of all different modes of reasoning to suit their needs.

However, we can talk about what sorts of modes of reasonings are depicted in the proofs that mathematicians provide to convey confidence in their ideas. In proof theory, we often see an operator known as "implies," typically represented by the turnstyle (⊢) which is a "metaoperator" in that no system defines the exact behavior of "implies." You're expected to understand what that means.

In a proof system, we set up a set of inferences which are deemed "acceptable for use" within the proof system in this way. For example, one might write {p, p→q}⊢q, which can be thought of informally to say "if p is true, and 'if p is true then q is true' then we can infer that q is also true." The list of these accepted inferences is typically provided up front. As a case study, one can look at First Order Logic, which defines a set of inferences in this way. Once the inference is accepted, it may be used in the proof from then on out.

So really, to answer your question, what we need to look at are the inference rules that mathematicians use. Many of these get phrased as axioms, so we can look there. If you look at the axioms we tend to use, most seem to be used to convey a deductive reasoning, but there are inductive examples as well. The one that comes to my mind is the Law of Induction, which is part of the Peano Axioms which define the natural numbers (0, 1, 2...). That law is very visibly designed to convey an inductive reasoning, to the point where they actually named the law "Induction!"

So mathematicians use both deduction and induction, along with whatever else is convenient to use. If you want to be more specific than that, simply look at the axioms which they use as part of their proofs, and you can see the reasoning they are trying to convey.

Answer 2

The inductive/deductive division that you are pointing at derives from a division of thinking which takes something outside of itself to be explained (in physics, the world), and that which takes itself to be a self-sufficient world (in mathematics, the axiomatic).

These are rough demarcations as they cross-cross each other; for example there is an axiomatic presentation of QM, which given the demarcation above ought to be mathematics, but is in fact physics.

In both, imagination is important; and is in a way an irreducible remainder, meaning that it is this what remains after all methods are exhausted (see Feyarabends After Method).

Here's an important new example: infinity toposes interpret Martin-Lofs intuitionistic type theory (this means it doesn't support the law of the excluded middle); when one looks at the development of this notion, it is a tangle, a thicket, a forest of many different intuitions and notions which aren't readily shaken into the simple demarcation you're suggesting.

Answer 3

I am aware of two major modes of reasoning: deductive and inductive. Whereas physics relies on induction, mathematics seems to rely exclusively on deductive inference, which ensures the soundness of its conclusions. Is that correct? Are there other modes of inference used in mathematics?

Deduction is a way of arguing about ideas that has some limited applicability to specific kinds of problems. Induction of the sort that is supposed to be used in physics is nonsense:

Deduction vs Induction -- are they equally valid?

You want to know what sort of justification is possible in maths. The answer is that justification is impossible, for reasons explained here:

Do all epistemologies suffer from the "regress of justifications" problem?

Mathematical knowledge, like all other kinds of knowledge, is created by noticing problems, guessing solutions to those problems and then criticising the guesses. Guesses in some parts of maths, like parts of number theory, are fairly precise and axiomatised and in those cases, the results can be proposed and criticised using deduction. Other parts of maths are less precise, e.g. - applied maths used by physicists.

All maths is done using physical objects: pens, paper, computers, mathematicians' brains and that sort of thing. As such, our knowledge of maths is dependent on our knowledge of physics since we have to guess that these objects are modelling the maths we are interested in. Our knowledge of maths is restricted to a large extent by the laws of physics, see "The Fabric of Reality" by David Deutsch, chapters 6 and 10, and "The Beginning of Infinity" by Deutsch Chapter 8.