Because all the induction methods we have seen so far are dangerous, induction is dangerous.
If you do not agree with the above argument, then congratulations, you have found an example where induction is unreliable, so induction is dangerous.
That’s all for now.
“Showing a Picture”
This calls for displaying a picture that was seen:
Inductive reasoning: a speculative approach.
Because induction (excluding mathematical induction) does not have logical necessity, it is essentially a kind of speculation.
Because of the proliferation of “spiritual drugs”.
Because of the rampant spread of “spiritual drugs”.
The Importance of Induction in Natural Science
The word “danger” is a relative term.
For example, if you are 170 centimeters tall and fall into water that is 180 centimeters deep, that is dangerous (assuming you cannot swim). But for Yao Ming, it is not dangerous.
In terms of reliability, induction is not as good as deduction, but it is stronger than analogy.
In fact, the foundation of “natural science” in human existence is induction.
Turning in Circles
Because spinning in circles in the same place
The Inductive Method
Your Inductive Reasoning:
(Red apples are edible, yellow apples are edible, green apples are edible) => (All apples are edible)
The True Inductive Reasoning:
Prove that
holds true.
Assume
holds true, and deduce that
holds true, thus proving the proposition.
Do you see the difference?
That’s right, the difference lies in a recursive process.
The incorrect inductive reasoning is similar to proving that F(1) holds true, F(2) holds true, F(3) holds true, and therefore, for all natural numbers N, F(N) holds true. Since natural numbers are infinite, you cannot list all possible variable values, so it is impossible to determine if there are exceptional values that make the proposition false.
Only by proving the validity of the recursive formula can you extend it from 1 to infinity. This is a rigorous method.
By the way, the foundation of mathematical induction is the Peano axioms.
The Peano axioms are as follows: (from Wikipedia)
0 is a natural number;
For every natural number a, there exists a unique natural number a' which is its successor;
For any natural numbers b and c, b=c if and only if the successor of b equals the successor of c;
0 is not the successor of any natural number;
For any proposition about natural numbers, if it is true for 0 and assuming it is true for a implies it is true for a', then it is true for all natural numbers.
归纳法可能导致错误的结论 - Inductive reasoning may lead to incorrect conclusions.
I don’t know who said it’s dangerous.
But saying it this way may be because the results of the inductive method may not always be effective, so there is a potential danger of drawing incorrect conclusions based on the results obtained.
For example, someone has seen 8789 apples of different occasions in their lifetime, and they were all red. So, they conclude that apples are always red. (Note: In fact, there are apples of other colors.)
If this conclusion is referenced, and in a crime scene the person does not see any red objects, they may conclude that there are no apples in this scene and relay it to others. Thus, this conclusion brings about a serious and subtle danger due to the inductive method.
Danger of Infinity
The Law of Natural Induction will bring infinite possibilities.
Infinite itself is already very dangerous.
Constructing Infinity through Infinite Progression
Induction Method
The induction method allows us to construct a concrete infinity through a progressive infinity (a virtual infinity).