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31 July - 6 August

21 August - 27 August

28 August - 3 September

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18 September - 24 September

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**Every now and then, I, the instructor, will want to say some thing very general to all of you, or pass on some message. This is where I will put them down. Visit every week.**

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**July 26,2012**. In the last lecture I started talking about the notion of space. We shall continue with that. Do give some thought to the topic.

By the way, I started off yesterday's lecture by mentioning to you a claim that logicians make:

**P**

*and*

**can not both be true simultaneously. I recall Bertrand Russell saying at one place in one of his essays that if**

*not*P**P**and

**are both true then every proposition is true. Well, neat and smart as his argument was, I do not recall what it was. All the same, surfing on the net, I found an interesting argument for a proof, which I paraphrase below;**

*not*PConsider the three propositions:

**P**: I am a frog

**Q**: I am not a frog

**S**: You are a goat.

Let us say we accept both

**P**and

**Q**to be true. Now consider the proposition

**P**. Since P is true,

*or*S**P**is true (That is in the nature of the or connective, if you remember.) Accepting that

*or*S**P**is true, now recall that, by hypothesis,

*or*S**P**is also assumed to be false (which is the same as saying that

**Q**is true). In that case

**S**is true! You are indeed a goat! Note that the argument holds for any proposition in place of S. Thus, given that

**P**and

**both are true then any arbitrary proposition is true.**

*not*PThese logicians! Well on a serious note, if logic interests you, you will find the slim little book

**Logic as Algebra**(by Steven and Halmos) very instructive, engaging and refreshing. You will admittedly need o first build a background in basics of logic before you are able to fully appreciate the contents. But it is worth giving it a cursory reading even without that.

**August 7, 2012**: Off and on, I will be discussing the notion of measurement, and its pitfalls. The following passage highlights in a dramatic way, the limitations of measurement. (It is taken from Hard Times by Charles Dickens (Bantom Books, Classic Edition, 1981; 823.8DIC]):

*"Thomas Gradgrind, sir. A man of realities. A man of facts*

and calculations. A man who proceeds on the principle that

two and two make four, and nothing over, and who is not

to be talked into allowing for any thing over. Thomas Gradgrind,

sir—peremptorily Thomas—Thomas Gradgrind. With

a rule and a pair of scales, and the multiplication table always

in his pocket, sir, ready to weigh and measure any

parcel of human nature, and tell you exactly what it comes

to. It is a mere question of figures, a case of simple arithmetic.

You might hope to get some other nonsensical belief

into the head of George Gradgrind, or John Gradgrind, or

Joseph Gradgrind (all suppositions, non–existent persons),

but into the head of Thomas Gradgrind, no, sir! (p. 2)"

and calculations. A man who proceeds on the principle that

two and two make four, and nothing over, and who is not

to be talked into allowing for any thing over. Thomas Gradgrind,

sir—peremptorily Thomas—Thomas Gradgrind. With

a rule and a pair of scales, and the multiplication table always

in his pocket, sir, ready to weigh and measure any

parcel of human nature, and tell you exactly what it comes

to. It is a mere question of figures, a case of simple arithmetic.

You might hope to get some other nonsensical belief

into the head of George Gradgrind, or John Gradgrind, or

Joseph Gradgrind (all suppositions, non–existent persons),

but into the head of Thomas Gradgrind, no, sir! (p. 2)"

**August 17, 2012**: We have started on vector spaces now. There is a very wide variety of good books for beginners. The viewpoint adopted by me will be very close to the one adopted by Axler [9] in his book, "Linear Algebra Done Right".(There are no discussion topics yet in this forum)