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

3 answers · posted 3y ago by Chgg Clou‭ · closed 3y ago by Peter Taylor‭

Question induction

#4: Question closed by $user avatar$ Peter Taylor‭ · 2021-12-21T14:06:20Z (almost 3 years ago)

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#3: Post edited by $user avatar$ Chgg Clou‭ · 2021-12-21T00:44:05Z (almost 3 years ago)

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By induction, I mean [this screenshot from this Youtube video](https://www.youtube.com/watch?v=-nn3XMoPC7s)
![](https://i.stack.imgur.com/s8VJy.png)
[Abduction as an Aspect of Retroduction | Chiasson, Phyllis | Commens](http://www.commens.org/encyclopedia/article/chiasson-phyllis-abduction-aspect-retroduction)
> _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.
[1]: https://i.stack.imgur.com/xHttZ.jpg

By induction, I mean [this screenshot from this Youtube video](https://www.youtube.com/watch?v=-nn3XMoPC7s)
![](https://i.stack.imgur.com/s8VJy.png)
[Abduction as an Aspect of Retroduction | Chiasson, Phyllis | Commens](http://www.commens.org/encyclopedia/article/chiasson-phyllis-abduction-aspect-retroduction)
> _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.
[1]: https://i.stack.imgur.com/xHttZ.jpg

#2: Post edited by $user avatar$ Chgg Clou‭ · 2021-12-21T00:43:37Z (almost 3 years ago)

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By induction, I mean [this screenshot from this Youtube video](https://www.youtube.com/watch?v=-nn3XMoPC7s)
![](https://i.stack.imgur.com/s8VJy.png)
[Abduction as an Aspect of Retroduction | Chiasson, Phyllis | Commens](http://www.commens.org/encyclopedia/article/chiasson-phyllis-abduction-aspect-retroduction)
> _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.
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/xHttZ.jpg

By induction, I mean [this screenshot from this Youtube video](https://www.youtube.com/watch?v=-nn3XMoPC7s)
![](https://i.stack.imgur.com/s8VJy.png)
[Abduction as an Aspect of Retroduction | Chiasson, Phyllis | Commens](http://www.commens.org/encyclopedia/article/chiasson-phyllis-abduction-aspect-retroduction)
> _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.
[1]: https://i.stack.imgur.com/xHttZ.jpg

#1: Initial revision by $user avatar$ Chgg Clou‭ · 2021-12-21T00:24:22Z (almost 3 years ago)

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Is Mathematical Induction truly "induction", or misnamed?

By induction, I mean [this screenshot from this Youtube video](https://www.youtube.com/watch?v=-nn3XMoPC7s)

![](https://i.stack.imgur.com/s8VJy.png)

[Abduction as an Aspect of Retroduction | Chiasson, Phyllis | Commens](http://www.commens.org/encyclopedia/article/chiasson-phyllis-abduction-aspect-retroduction)

> _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.             
&nbsp; &nbsp;  &nbsp;  &nbsp; 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]). 
&nbsp; &nbsp;  &nbsp;  &nbsp; 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. 

[![enter image description here][1]][1]


  [1]: https://i.stack.imgur.com/xHttZ.jpg

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