Pi

Question - Give me a method of constructing a line segment of length Pi.

:-)

Defying Logic

For time immemorial, I have believed in the study of Fundamentals. Whenever the study or analysis of any topic of subject is considered, I have always believed that going to the fundamentals will always yield fruits.

After being a student of Mathematics for sometime, I found myself gradually drifting towards areas which can be roughly classified as Mathematical Logic and Scientific Philosophy. These areas of studies, fundamental as they are, held an immediate appeal for me. For sometime, I even thought these were the only areas that would interest me.

I should admit that inspite of my interest, my knowledge in these areas still is extremely limited. But the fascination remained with me for quite some time.

Of late, I have started to drift from that position. I am giving an apology here for my change of stance. There are a few reasons.

Even though, I still believe these fundamental studies are important, I think they are more for the past-masters of Mathematics. These fundamental studies require not just intellectual power, but also a vast experience in Mathematics. Of course, intellectual power also is important for deep fundamental study but they require some level of overall knowledge of what the upper-layer(which is mathematics) looks like. To gain that, considerable amount of time and effort should be spent on atleast a few major areas of Math which can only be done if you are a full-time Mathematician for some period in your life.

Mathematics - whatever form it might take - is always a study of structure and pattern. Infact the concept of "abstraction", which forms the core of higher mathematics, is all about pattern-searching and as a result Generalizing. This is where I find myself in conflict when trying to proceed with Mathematical Logic. Logic forces me to ask questions on my basic assumptions. It tries to break all pattern by trying to reach to the bare fundamentals. So much so that we start doubting our very existence many times.

Anything said in the last paragraph about logic applies for Scientific Philosophy - the art of investigating truth by asking never-ending questions.

After spending some time over such fundamental studies, I have come to the conclusion that - even though these studies are important and have helped historic breakthroughs in the human thought process, they are not suitable for students of mathematics. Students of Mathematics should rather delve into Real Mathematics - which is the study of pattern and structure and the creation of new patterns and structures.

There is a vast ocean of Mathematical knowledge which can be so much fun to experience. Once we have experienced a significant portion of it, we may choose to drift into more fundamental studies of logic and philosophy. However, to start here and go beyond, is like trying to learn the working of a BMW engine rather than taking off on an open highway to explore newer worlds!

Logic as a matter-o-taste

It has been sometime since I have thought about these things. Here is a little argument about null sets that I want to present. Please find flaws.

If A and B are two sets such that 'x' belongs to A implies 'x' belongs to B, then A is a subset of B. The null set is a set that does not contain any element. Let us look at the following statement which is generally held to be true in Set Theory.

Statement S --> The null set is a subset of every set.

The argument used (or is hidden) to support this is as follows. In the first part of the definition of a subset, ("x belongs to A") since finding an x belonging to A, when A is the null set, is impossible - we say that every x belonging to the null set belongs to any other set.

In other words, we are saying that if at all there is an x belonging to the null set, it would belong to any other set. But since there is no such x, we can safely say that the null set is a subset of every other set (including itself).

On the contrary, equally logically, we can argue that the statement S is not a verifiable statement because the definition of subset needs an element x.

Thus, I claim that the accepting statement S as true is just a logical choice. It is not a 'primitive' concept which would be evidently true.

In more mathematical terms, if we make the choice of accepting that S is not verifiable, then the partial order relation "is a subset of" cannot be defined over any collection of sets that has the null set as one of its members.

Brahmins in South India

The Kanchi Seer has said that Brahmins are (and were) part and parcel of Dravidian culture always and they are not immigrants like some Dravidian Movements (and others) claim and believe.

I don't know how much of this is based on credible historical facts - but am sure interested to know how far we can go back in time on this issue with acceptable (close to fact) theories.

Music musings

I wonder what fundamental property of our brain makes us appreciate musical tunes (or raagaas).

How is it that there seems to be an almost definite distinction between what is in-tune and what is out-of-tune (abaswaram)?

Assuming that SOUND is a continuum - extending from negative infinity (base tune) to positive infinity (high pitch) - is there an analytic definition of what sequences of sounds will constitute "in-tune" and what constitutes "out-of-tune"?

And what is it that a Musical System - like say Carnatic - is trying to achieve by binding itself into rules. Is it just trying to operate on a restricted (discrete) domain instead of the whole of the Sound Continuum? If yes, what is the point of doing that?

I tried looking at the definition of Raaga (in Carnatic terms) as a function F that maps the finite set (1, 2, 3... n) into the Sound Continuum with a given set of rules. For eg., one rule might be -

--> F(x) always belongs to one of (Sa, R1, R2, ..... , Ni2) in some scale and not to intermediate sounds. Like the sound between Sa and Ri1.

I am in the process of building this definition into a more complete form. May be an analytic definition already exists and is well known. But I am still on the look out for one and when I am in that process I wont keep myself from trying to build my own definitions. It is fun trying to make these things analytic.

Please pool in your ideas if any. I have just drafted my own without actually trying to spend time in bringing them to any kind of final form! :-)

In other news...
Humour and sensationalism can never be substitutes for good writing. This truth was reiterated recently in two pieces on cricket in Rediff.

While Sriram Ranganathan thought that mocking at Australia by calling them whatever he feels like after one test defeat will earn him good audience, Salil Kumar chose the less-sensational path of reminding us all that at the end of the day cricket is just a game.

The latter's writing is worth reflecting upon for it lends a sense of balance while the former's writing reminds me of what Rajaji once reportedly commented about a fellow speaker -

"..(his speech was) a diarrhoea of words and a constipation of thoughts!"

Full of senseless criticism and personal attacks, most parts of the article can be dismissed as sheer rubbish. I hope not many future writers would get inspired by this kind of writing style.

Thankfully, rediff had a disclaimer at the end of the article. So much, for guest writers!

Rumsfeld's tough-talking recognized..
So it is not just me..

I talked a bit about the special skills that US Defense secretary Donald Rumsfeld possessed when it comes to public speaking!

And I thought it would go unnoticed - but no!

Somebody has recognized it and given him an award.

Other recipients of the awards include California governor elect Arnold Schwarzenegger! :-)

Macgill on Sachin..
Dear Mr. Macgill,
Now that you have said this, you should better check on what another spinner from your land has to say about Sachin.

And while you do that, bear in mind -
1. It is quite a debate the world over on whether Shane Warne is *the best spinner ever* in cricketing history.
1a. I have not heard anything even remotely close to that about one Mr. Stuart Macgill. Rather, I am mostly left wondering what Channel Nine commentators get out of hyping Macgill up - inspite of having seen someone like Warne. Maybe they have a flair for advertisement!
2. Shane Warne has played(bowled to) Sachin Tendulkar in 11 test matches spanning over 4 seasons. The whole world watched the clash and the results are well known. Not to say anything about the one-day clashes in Sharjah!
2a. Stuart Macgill made it to the India Tour once and played ONE tour game. Maybe the Australian think tank thought he was too good a talent to waste on Sachin Tendulkar in Test Matches.

So, I do know all of the above facts. And knowing all this, you know what I think about your recent talk.

If you had said all that you have said after returning figures atleast as good as those of Warne's against Tendulkar, I would have let it go!

Trivia..
Let me play quiz master for a bit.. Here is a trivia that I came across recently.. And I have tried to rephrase it to make it sound difficult..

As a result of which, the question might sound really awkward.. :-)

In the topic of National Anthems, India has something in common with one other country. And this common thing is enjoyed by only these two countries. What am I talking about?

Inimitable Vir Sanghvi on Jayalalitha..
Read Vir Sanghvi on Jayalalitha's latest "arrest-all-press-so-no-one-writes-about-you" drama..

I would like to quote two portions.. ,

Quote 1
"...(During an interview) Her view of herself was as a victim; her view of the world was that it was her enemy; and her way of coping was to keep fighting the world because otherwise it would victimise her again.

Ever since her comeback began, she has behaved almost exactly as I feared. "

Quote 2
..Can India really afford a deeply disturbed person as chief minister of one of its most important states? Or is it time that we all put our heads together and found her a nice room in a comfortable sanatorium somewhere?..