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