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Q&A

Is Mathematical Induction truly "induction", or misnamed? [closed]

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Closed as off topic by Peter Taylor‭ on Dec 21, 2021 at 14:06

This question is not within the scope of Mathematics.

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By induction, I mean this screenshot from this Youtube video

Abduction as an Aspect of Retroduction | Chiasson, Phyllis | Commens

Induction:
The prefix “in,” also from the Latin has to do with inclusion. Thus, the prefix “in” (to include) combined with the suffix “ductive” means “leading into” (or including), as one would do when reaching a conclusion by estimating from a sample, or generalizing from a number of instances.

Therefore, based upon their Latin derivations (to which Peirce was partial, as he was for Greek roots) our four terms have the following meanings:

  • Retroduction = deliberately leading backward.

  • Abduction = leading away from

  • Deduction = leading to separation, removal, or negation.

  • Induction = “leading into” (or including).

The following quotation moots Deduction and Induction — not Abduction — but it doesn't expatiate or address whether Mathematical Induction is correctly named?

1.1 Introduction

In the sciences and in philosophy, essentially two types of inference are used, deductive and inductive. Deductive inference is usually based on the strict rules of logic and in most settings, deductive logic is irrefutable. Inductive reasoning is the act of guessing a pattern or rule or predicting future behavior based on past experience. For example, for the average person, the sun has risen every day of that person’s life; it might seem safe to then conclude that the sun will rise again tomorrow. However, one can not prove beyond a shadow of a doubt that the sun will rise tomorrow. There may be a certain set of circumstances that prevent the sun rising tomorrow.
        Guessing a larger pattern based upon smaller patterns in observations is called empirical induction. (See Chapter 6 for more on empirical induction.) Proving that the larger pattern always holds is another matter. For example, after a number of experiments with force, one might conclude that Newton’s second “law” of motion $f = ma$ holds; nobody actually proved that $f = ma$ always holds, and in fact, this “law” has recently been shown to be flawed (see nearly any modern text in physics, e.g., [56, p.76]).
        Another type of induction is more reliable: Mathematical induction is a form of reasoning that proves, without a doubt, some particular rule or pattern, usually infinite. The process of mathematical induction uses two steps. The first step is the “base step”: some simple cases are established. The second step is called the “induction step”, and usually involves showing that an arbitrary large example follows logically from a slightly smaller pattern. Observations or patterns proved by mathematical induction share the veracity or assurance of those statements proved by deductive logic. The validity of a proof by mathematical induction follows from basic axioms regarding positive integers (see Chapter 2 for more on the foundations of the theory).

David Gunderson, Handbook of Mathematical Induction (2010), pp 1-2.

3.1 The first principle

For convenience, the standard presentation of mathematical induction is repeated here. Sometimes this standard version of induction is called the “first principle of mathematical induction”, and is also called “weak mathematical induction” (as opposed to “strong” induction, a modification appearing in Section 3.2). Recall that the notation P → Q is short for “P implies Q”.

Theorem 2.4.1 [Principle of Mathematical Induction (MI)]

Let S(n) be a statement involving n. If
$\qquad$ (i) S(1) holds, and
$\qquad$ (ii) for every k ≥ 1, S(k) → S(k + 1),
then for every n ≥ 1, the statement S(n) holds.

Op. cit. p 35.

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3 answers

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Well there are many ways to look at this, but I would argue yes.

To use your last screenshot: the cause is S(k), the effect is S(k+1), and it's up to you to fill in how you get from one to the other.

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In the context of Peano arithmetics, mathematical induction is actually an axiom, that is you cannot prove it from other properties of the natural numbers. It is extremely plausible that if something holds for $0$, and if it holds for $n$ it also holds for $n+1$, that it holds for all natural numbers. However the only reason why you can use it in proofs is that you have it as axiom. In other words, it is assumed that it works. Therefore I'd argue that it is justified to call it “induction”.

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Mathematical induction is a case of deductive reasoning.

It uses some kind of inductive principle, and the justifications for why that holds are a different matter. But its use is a case of rigorous, deductive reasoning.

Finding the matter to be proved deductively is often a creative process that involves induction. This is also true for mathematical induction; finding the rule to be established might be a process of induction, which is then verified by deduction.

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