Induction (again)

Prof. K.M. Tsang   10:26 AM (3 hours ago)           Thank you for your email and I am glad that you still remember what I have said in my lectures almost 20 years ago.     The most widely accepted formulation for the system of natural numbers is the system of the Peano Axioms.  You can read a brief description of this from the following link http://global.britannica.com/EBchecked/topic/447921/Peano-axioms       As you can see, the fifth axiom here is essentially the principle of mathematical induction.     I already forgot why I spent five minutes to emphasize the difference between "mathematical induction" and the "principle of mathematical induction".  There should be no or little material difference between them, except that we (and students) should understand that mathematical induction is one of the axioms (in the Peano's axiomatic system) for the natural number system, not to confuse it as "a theorem". Hence it is called a "principle".  We can use it to prove many assertions concerning validity on natural numbers.  This is also (logically) equivalent to the well-ordering property of the natural numbers (under the usual order in the system of the natural numbers).     I hope my explanation can be of use to you in your classes.  Let me know if you need further clarifications.     With best wishes,     KM Tsang

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An interesting question in Mathematical Induction 2 messages

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	Mon, Jun 8, 2015 at 6:43 PM
To: kmtsang [AT] maths.hku.hk
Dear Prof. Tsang,   I was your HKU student of the course "24114 Linear Algebra" in 1996-1997, many years ago when I was in University Year 1. I have an algebra question that would like to discuss with you (or, to seek your opinion).   What is the difference between "mathematical induction" and "the principle of mathematical induction"? This question sounds rather philosophical, yet I remember at that time, you spend over 5 minutes in your lecture, talking about their difference. And your conclusion is that we should use "the principle of mathematical induction".   However, to be honest, at that time I don't really understand. I just memorize that I should use the phrase "by the principle of induction", and then I tell myself to avoid teaching my students "by induction". Although you know that nowadays HKDSE Maths Module 2 exam marking scheme is not very strict and they even accept phrase like "by M.I.", I still want to teach my students the correct phrase.   So, would you mind explaining it again to me, or is there any reference that I can read myself?   Thank you for your attention!

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Prof. K.M. Tsang	Tue, Jun 9, 2015 at 10:26 AM
Thank you for your email and I am glad that you still remember what I have said in my lectures almost 20 years ago.    The most widely accepted formulation for the system of natural numbers is the system of the Peano Axioms. You can read a brief description of this from the following link http://global.britannica.com/EBchecked/topic/447921/Peano-axioms    As you can see, the fifth axiom here is essentially the principle of mathematical induction.    I already forgot why I spent five minutes to emphasize the difference between "mathematical induction" and the "principle of mathematical induction". There should be no or little material difference between them, except that we (and students) should understand that mathematical induction is one of the axioms (in the Peano's axiomatic system) for the natural number system, not to confuse it as "a theorem". Hence it is called a "principle". We can use it to prove many assertions concerning validity on natural numbers. This is also (logically) equivalent to the well-ordering property of the natural numbers (under the usual order in the system of the natural numbers).    I hope my explanation can be of use to you in your classes. Let me know if you need further clarifications.    With best wishes,    KM Tsang

 

 

 

 

 

Induction

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我係從 logic 角度看 MI 這個問題:
- proposition 絕不等同 statement, P() 肯定只係 proposition, P(k) 才是
statement; 換句話講, 我認為教科書與老師不該使用 statement, 以正視聽.
- 但會考中無必要為考生使用 proposition 還是 statement 作評分考量, 亦無必
要為考生是否串錯 proposition 一詞作評分考量, 因為這只是語文能力問題.
- 但如果考生錯用使用 P(k), 這不是單純是語文能力問題, 大多數情況都顯示考
生對數學歸納法的原理有所不明, 只有死背數學歸納法的機械操作. 最明顯的錯誤
係寫 P(k)= k/6 (k+1)(2k+1), 應予零分, 或在只有局部犯錯下批予小量步驟分.

WONG Wai-chi, Norme wrote:
> 那位老師肯定未改過會考卷，兼且不知考試局一早說這樣寫是錯的。
> 不過，就算中大那本教科書也是這樣寫。
>
> 1.利用 P(k) 的寫法，錯多過對。不建議使用。
> 2.proposition 多數串錯，不如簡單的寫 statement
> 3.假設 n=k 時是對的：Assume the statement is true for some positive integer k,