Handout Zero: Memorandum of Understanding File 220.4KB PDF document
What is the course about? What is expected from the students? What are the expectations of the students from a course like this one on mathematics? I feel that for issues such as these to be meaningfully examined, the stage is best set by agreeing to some kind of broad mutual understanding right at the start of the course. "Memorandum of Understanding" as I have called this document, is meant to do just that. I hope it does.
This piece attempts to lay down the basic idea that has led to the introduction of this course.
A very good introductory presention of the basic ideas and history of representational measurement theory.
In addition to his seminal monograph, `Algebra of Probable Inference', this paper of his is an equally important contribution in the theory
of probability. Relate what i have tried to convey in the lectures to the ideas put forth here.
A very lucid and readable presentation of the significance of Cox's approach in the general context of characterizing measurable attributes of what we perceive of the world.
A good paper that should be read with Cox's book.
Boolean algebra is commonly understood to have many separate applications. It is used in the design of logic circuits. It is the backbone of probability theory, and so on. It is time that you looked at boolean algebra within a broader framework, that of posets and lattices. More on this in the lectures. For the present try to absorb what Knuth has to say.
Related to logic, boolean algebra, probability theory, this is another of Kevin Knuth's (Do not confuse him with Donald Knuth) very engaging paper. Do take a look.
An interesting discussion on how to structure proofs. I first noticed this paper in 1985.
Could we make proofs more human? Proofs are generally terrifying, and the thought of having to go through them unpleasant. The way they are commonly presented, they seem to be rigid and unresponsive to the readers queries, cold and take-it-or-leave-it stuff. On the whole machine-like.
Turning to logicians is of no help. Their writings have become far too specialized to be easily followed by nonlogicians.
So, can we on our own do something to ease the situation, at least for the purposes of teaching? Could we make proofs more `human', combining pleasure with learning? Are there some general guidelines that we could follow in this endeavour? Leron's suggestions might be of help.
A `tongue-in-cheek' slogan: Rigour dehumanizes, and absolute rigour dehumanizes absolutely'
In talking about a system as a black box, I mentioned in one of the lectures that the idea is linked with the philosophical position known as pragmatism. This paper expands this viewpoint.
An ineresting set of notes on the Peano Postulates. I can not locate the authour. Several subtle issues are discussed. To be read at liesure
The snapshot shows an example of a square subdivided into (26) squares no two of which are of the same size. It also shows a colouring scheme for the squares such that no two adjacent ones have the same colour. How to carry out such a subdivision? Well look at the papers [1] and [2] for an interesting approach based on an a reformulation of the problem as an electrical circuit problem
The colouring scheme has to do with what was until not very long ago called `the four-colour conjecture': you need at most four colours to colour any map such that no two adjacent regions have the same colour. Look up what it is, and is it still a conjecture? What is in the mathematical parlance called a conjecture?
Weekly outline
13 August - 19 August
As I have tried to convey to you in my Handout Zero, writing is an integral part of your daily activities.These notes are very helpful in this connection. I am sure you are familiar with the pioneering contributions of Knuth. If you are not, then do look him up on the net.
This is a pdf version of the hand written notes with the same title.
Aside: Music of the Week: I recommend Vishwa Mohan Bhatt on Guitar, or rather on Mohan veena (a hybrid of guitar and veena,
innovated by him). You can start with his youtube recordings, titled `Galaxy of Strings', which starts off with raga bageshari,
and hansdhwani.
20 August - 26 August
27 August - 2 September
3 September - 9 September
10 September - 16 September
17 September - 23 September
24 September - 30 September
1 October - 7 October
8 October - 14 October
15 October - 21 October
22 October - 28 October
29 October - 4 November
5 November - 11 November
12 November - 18 November
19 November - 25 November
26 November - 2 December