How does abstraction/generalization in mathematics fit into inductive reasoning?

Question

I have a question about the nature of generalization and abstraction. Human reasoning is commonly split up into two categories: deductive and inductive reasoning. Are all instances of generalization examples of inductive reasoning? If so, does this mean that if you have a deep enough understanding of inductive reasoning, you broadly create "better" abstractions?

For example, generalizing the integers to the rationals satisfies a couple of things: the theoretical need to remove previous restrictions on the operations of subtraction and division, and AFAIK the practical need of representing measurable quantities. This generalization doesn't seem to fit into the examples given here http://en.wikipedia.org/wiki/Inductive_reasoning at first glance, and I was hoping someone could give me some nuggets of insight about this. Or, can someone point out what the evidence is that leads to this inductive conclusion/generalization?

Answer 1

You're correct that moving from the integers to the rationals does not fit, because the generalisation that inductive reasoning refers to is a generalisation of statements (or predicates) not of the objects themselves.

For example, you could generalise from the statement "all the even numbers above 3 we ever tried can be written as the sum of two primes" to "all of them can" - and that's an example of inductive reasoning. (There's there's no known deductive proof of this conjecture.)

Even numbers can be "generalised" to all numbers, but that's different to inductive reasoning. We don't move by inductive reasoning to "all whole numbers above 3 are the sum of two primes" because we find that 11 doesn't work.

Generalising generally vs inductive reasoning

"Generalising" the integers to the rationals is a superset relationship, which I can write very simply in maths notation, because it's like

The generalisation that inductive reasoning makes is:

• 
• 
  hence we believe that
• 

I've not generalised A to B, I've generalised P from being true in all of A to being true throughout B.

Better abstractions

In fact, you could say that the rationals are an example of a field, whereas the integers are only an example of a ring - a generalisation of a field. Mathematicians "generalised" (not inductive reasoning) from the number systems, matrices and other examples to make the abstractions of groups, rings and fields. Some theorems about fields can be generalised to rings, which would be an example of inductive reasoning were it not for the fact that mathematicians don't admit inductive reasoning as proof, and proved these generalisations deductively!

(Proof by induction is, perhaps confusingly, an example of deductive reasoning, but is often something you do after having used inductive reasoning to conclude that you might want to prove your statement.)

A deep understanding of inductive reasoning isn't necessary for good abstractions, but good abstractions result from deep understanding of diverse examples.

Summary

Abstraction is a form of generalisation, and supersets are a form of generalisation, but inductive reasoning specifically means generalising statements from examples to every example.

All inductive reasoning is generalisation, but not all generalisation is inductive reasoning.

Answer 2

Induction in the sense that philosophers talk about it is the idea that ideas are derived from observations in some sense and can be proven or made more probable by observations. Induction in the philosopher's sense is impossible. (I will discuss mathematical induction below.) Explanations do not follow from observations in any sense. Nor do observations prove any idea. Nor can any observation make any idea one jot more probable. Inductivism is just another variety of justificationism: the idea that it is possible and desirable to prove ideas true or proably true. In reality, you can't prove any position or show it is probable. Any argument requires premises and rules of inference and it doesn't prove (or make probable) those premises or rules of inference. If you're going to say they're self evident then you are acting in a dogmatic manner that will prevent you from spotting some mistakes. If you don't say they are self evident then you would have to prove those premises and rules of inference by another argument that would bring up a similar problem with respect to its premises and rules of inference.

In reality all knowledge is created by conjecture and criticism. You notice a problem with your current ideas, propose solutions, criticise the solutions until only one is left and then find a new problem. Experiments are useful only as criticism. Ideas can't be derived from experiment any more than from any other set of premises. Rather, the idea is that you work out how the consequences of one theory differ from those of another. Then you conjecture ideas about experimental setups that would enable you to see the relevant consequences and criticise them. Once you have a setup that works about as well as you can make it work you use it to do the test. If the results are compatible with one theory and not the others then you may have successfully refuted some false ideas. Sometimes a purported successful experimental test will be successfully criticised because a test is a conjecture about something that happened and that conjecture may be wrong, so experiments don't prove anything.

What about deduction? Deduction is just working out the consequences of a conjecture. It may be used as part of a rational argument or an irrational attempt at argument. For example, if you start an argument by ignoring a criticism of your position, then the rest of the argument is just elaborating on something that may be a mistake: it is an irrational waste of time.

Mathematical induction is just a variety of deduction. For example, there is a proof of the idea that (sum from j = 1 to j = n of j) = n(n+1)/2. The proof usually runs like this. (1) This equation is true for n = 1. (2) If it is true for n we can argue that it should also be true for n+1. (3), (2) and (1) imply it is true for all n. This is very different from induction of the sort usually imagined by philosophers since the implication that the statement is true for all n is drawn using an explicit and very specific conjecture that we can argue about and that argument might lead to interesting new ideas.

Likewise the "generalisation" from the integers to the rationals is a result of proposing a conjecture to solve the problem of how to deal with cases in which you want to deal with quantities that don't have integer values. In other words it is an instance of knowledge creation by spotting a problem and proposing conjectures to solve that problem.

Books worth reading on this include "Realism and the Aim of Science" by Karl Popper, Chapter III (the title of that chapter is "Metaphysics: Sense or Nonsense?" but it discusses maths). "The Fabric of Reality" Chapter 10 discusses the creation of mathematical knowledge. See also "Proofs and Refutations" by Imre Lakatos (this book is good though a lot of what Lakatos wrote about epistemology after that was junk).