Through the passage of time, I am now a "senior member" of the academic community; roles like Professor, Head, Dean etc. are a consequence.
A leader would take such role to mould the community around her/him. A leader would be able to delegate much of the work to others while ensuring that all policy matters were referred back. A leader would lead the community into possibly risky but potentially fertile territory. A leader would also take the blame where the group fails.
The leadership interpretation of these roles is too demanding for me! I prefer to see these roles as service roles. You need a length of rope? Let me see if I can find some. You want suggestions on how to use it? Happy to give a discourse. Tied yourself into knots? Maybe I can help you get free.
I also see these as guidance roles. One has experience and the analytic skills to organise that experience into rough guidelines about what is likely to work. "Last time I went that way, there was a swamp with alligators in the path. Perhaps you need to carry a big spear or learn to traverse the treetops."
To a large extent, this difference percolates into one's teaching style. I would like to spread out the ingredients and demonstrate one way to make something tasty, rather than say "Here is a recipe."
Leaders like to see the world as made up of leaders and followers. Guides like to see the world as made up of guides and potential guides.
]]>Numbers play multiple roles:
For natural (finite) numbers, these different senses of the use of numbers coincide. (Though one can have some doubts when the numbers are really large!)
In the same way, mathematical objects can be considered in different ways:
Even for finite sets, these notions do not coincide! For example, there are many different finite groups of the same size.
Coming to the problem of "infinity". The simplest notions of infinity are:
Each of the above have associated arithmetic and algebraic operations. For example, with counting numbers we have addition and as a consequence multiplication. With ordinal numbers we have the notion of a successor which can be used to define a notion of addition. The corresponding structure in sets is that of Boolean or sigma algebra of sets. Category theory also has its own notion of algebra called ``universal algebra'', which is like (but not quite the same as) the sigma algebra of sets (infinite sums and products need to be defined and may not exist!).
So to re-phrase the question, we are asking if the ordinary notion of arithmetic and algebraic operations extends to infinity.
At first glance it does. We can certainly perform Boolean operations with infinite sets. The problem is that the usual statements about these operations are sometimes no longer true and our intuition about algebraic identities would fail us.
For example, it is usual to say that multiplication is the operation of repeated addition. When the number of additions is infinite, it is not very evident what this means. We define the product of sets AxB which clearly explains what this operation (multiplication) is for sets.
Similarly, it is natural to think of addition as repeated successor operations, but it is not always clear what this means for the infinite successor operation. Again, ordinal succession is defined in a way that such an operation is meaningful through the notion of limit ordinals.
However, in each case some "obvious" results from the finite case are no longer valid.
It is worthwhile to extend notions from the finite to the infinite when this is useful in giving us expectations regarding questions (about finite sets!) that we could not have arrived at otherwise. (For an interesting example, have a look at the Goodstein sequence.)
As an addendum, I would like to add the naive re-statement of Skolem-Lowenheim.
Since language consists of countably many sentences, we can only hope to define countably many things and from a practical point of view we can only define finitely many things.
Thus, infinity is a notion that mathematicians handle with care, limiting the roles that it can take, so that playing around with infinity gives meaningful (and correct!) results about the finitely many things that we will actually encounter! Mastering this way of handling infinity with care is what a lot of mathematical training is about.
[F]or anyone active in mathematical research today, most mathematical learning has happened outside the classroom.
This is not all that unusual in the sciences. In fact, in most experimental sciences, enormous amounts of training happens "on the job". In other words, one is learning by observing others "do" science and mimicking them. One gets lifted onto the "shoulders of giants" without having to climb all the way up there!
Now, this similarity may leave the wrong impression that mathematics is only learnt through "apprenticeship"; in other words that Ekalavyas are not possible in mathematics. By Ekalavya, I do not refer to the gruesome and disgusting part of the story that popular mythology is so fond of. Rather, I am struck by the idea of a student who is inspired to self-study in a remote land which has little or no local expertise. This too is non-classroom learning!
Ramanujan is one of the well-known Ekalavyas but there are many less famous examples. These are mathematicians who took up a subject that was relatively unknown in their geographic vicinity (in the era before modern communications) and read (and worked hard to understand!) whatever was available, to teach themselves, and ended up with a novel viewpoint not known to the remote specialists!
Such people are not common in mathematics; the point is that they exist! I think this is because mathematicians insist on precise communication. One goal of excellently written mathematics has always been that it should be possible for someone with access to the relevant literature (and adequate patience and intelligence) to verify its correctness without asking experts for clarifications.
In the sciences, it is almost impossible to make a dent without an early apprentice-ship. In fact, it always surprises me how different string-theorists are from mathematicians in this respect, considering that most physicist see them as "too mathematical". In fact, the sociology of doing string-theory is a lot like the experimental sciences even though the theory is still far removed from experiments!
]]>Linux on a netbook is like a young penguin exploring the Antarctic on an ice floe!
Just recently, I have had a chance to play with (and to use) no less than three different OS's designed for a netbook --- Ubuntu Netbook Remix, Moblin v2 and Sugar on a Stick. One of these has already replaced^{1} the Linpus Lite that came pre-installed on this Acer Aspire One (acquired by IMSc six months ago). However, it could easily have been one of the others; they are all (to use a teen-phrase) "awesome".
In each case the "installation instructions" consist of the following steps:
All of them installed with ease (in the It Just Works(TM) fashion) on the AAOne.
I provide a short description of each below but to really experience them you have to use them! As the Sugar folks might say you need to enjoy them as a child would.
Ubuntu Netbook Remix will feel the most familiar to those who have already used computers before and have certain pre-conceived ideas about how to use them. It still deviates from the norm:
Other than that it is Ubuntu under the hood (which is Debian under another hood; all of which makes it a bit like the mask removal scene in Mission Impossible). In fact, but for some very minimal eye-candy, it is surprising how similar this interface is to the ratpoison-like interface that I have been using (under different window managers!) for quite a while now. In particular, it can be driven from the keyboard.
Moblin v2 is currently a beta version but it is fully usable as-is on the AAOne. Of the three, one feels the Moblin guys have the most talented graphic artist. The boot-up, the background and the interface are all quite beautiful. The interface will be "strange" to people with pre-conceived notions (read "adults") but is really quite intuitive. The buttons are similar to the Sugar interface and are based on simple pictograms instead of the usual elaborately drawn icons.
However, it is really Sugar on a Stick that is the most amazing OS. First of all it allows one to work completely off a 1GB USB stick including keeping some persistent data on the stick (the more stick there is, the more data you can "stick" to it!). The colors and graphics are minimal, yet intuitive and beautiful. What makes it superior to the other two is the "activities" (which roughly but not exactly replace "applications") that are immediately available. This is not an adult's toy --- it is a child's toy. Adults have a very definite idea of what makes them happy and they want immediate (if not instant) gratification. A child enjoys every bit of the journey --- exploring the box that the toy came in, tinkering with the toy in ways that the designer didn't imagine, and so on. Most importantly, a child knows that the toy is more than "just a toy".
If you are a parent who is mildly familiar with computers, I urge you to follow the simple steps above and provide your child with a computer interface that belongs their generation instead of ours. Or, if you are an adult or teenager and the child inside you still lives, then provide that child with some joy by doing the same.
There is an unfinished draft in my blog directory with the title "Simplicity doesn't mean Dumbing Down". I probably don't need to write that essay anymore, I can just point to the Sugar interface.
Of course this does not affect my "work" platform which runs as a Debian stable chroot under any Linux-based OS; in particular, any of these will do. ↩