Numbers play multiple roles:

• As a way of counting. (Cardinals)
• As a way of ranking. (Ordinals)

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:

• "Pure" set-theory. (Axiomatic set theory)
• Sets with some sort of structure. (Category theory)

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:

• The set of natural numbers. (Cardinal aleph_0)
• The ordered set of natural numbers. (Ordinal small omega)
• Asymptotic points or points at infinity. (For example, the point (1:0:0) in projective geometry)

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However, in each case some "obvious" results from the finite case are no longer valid.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.

The kind of "popular mathematics" that interests me is more in the sense of "popular mechanics" or "popular electronics". Something that gives the reader new things to do and helps build tools to explore further. Just as an essay on Newton or Hook is not an article on popular mechanics, and an essay on Marconi or Baird is not popular electronics, so a book like that by E. T. Bell on the lives of famous (dare I say "great") mathematicians does not constitute "popular mathematics". (For example, this essay is not (yet!) popular mathematics.)

Another aspect of "popular mathematics" is that mathematics should be its own narrative. An article on "building your own rubber-band powered airplane" that spent a lot of time discussing the joys and benefits of flying, would not interest me. In the same sense, an article that extols the beautiful applications of what one is about to study is wasting time in getting to the point --- if it is beautiful it will speak for itself.

There is certainly some interest in how the ideas (might have) come about. A pure recipe, which says, do this calculation followed by this substitution to have this formula and "Voila!" is meant for computers --- not human beings. "How does (did) one think of the steps that are to be carried out?" is relevant to the story. This need not be a factual/historical account as long as it is plausible and interesting!

As an example, consider the following (brief) account as how one can build up the notion of complex numbers from geometry.

In the study of the geometry of a (straight) line, one encounters numbers via translations to the left and right (traditionally, positive and negative respectively) which corresponds to addition of numbers. One also encounters scaling up or scaling down which corresponds to multiplication and division. What happens when one considers that plane?

One can certainly translate the plane in various directions. We can also rotate the plane and scale it. The recognition, that (after fixing a centre/origin and a unit "x" direction), each point in the plane can determine a rotation+scaling as well as a translation, allows us to define a multiplication as well as an addition of points on the plane. This operation creates the arithmetic operations making up the complex numbers.

One can also follow this up to see how regular polygons correspond to cyclotomic numbers and how constructible numbers involve solutions of quadratic equations. This can lead to a derivation of the recipe for the construction of a regular pentagon --- something which is rarely done in high school geometry books.

There are number of good books on popular mathematics:

• Mathematics for the Millions, by Lancelot Hobgen
• Figures for Fun, by Yakov Perelman
• Geometry and the Imagination, Hilbert and Cohn-Vossen.
• Mathematician's Delight, by W. W. Sawyer
• Godel, Escher, Bach, by D. Hofstader

There are a number of good books which are not popular mathematics:

• The Man Who Knew Infinity, by R. Kanigel
• The Men of Mathematics, by E. T. Bell
• The Code Book, by Simon Singh
• Fermat's Last Theorem, by Simon Singh

Too often (like in the current case!), one starts with the noble goal of writing popular mathematics, only to succumb to the (somewhat lazier) approach of writing about mathematical philosophy or mathematicians. This is not the only time I have done so! With faint heart, I recall an article that appeared in Science Age in 1984 on the work of Gerd Faltings --- an excellent article by Madhav Nori was vandalised by yours truly under pressure from the editors to make it "more accessible"!

In response to a query from Dr. R. Ramachandran who writes about Science and Science Policy for ``The Hindu'' newspaper group, I wrote the following short piece which others may find interesting as well.

Milnor is well-known for his work on topology and geometry. He has also made significant contributions to algebra and even number theory. In addition, he has written a number of books which are loved by graduate students in Mathematics all over the world.

One of the striking early results of Milnor was the example he gave of a seven dimensional space which is topologically a sphere but its geometric (differentiable) structure is different. This was the first example of an "exotic sphere". A nice way to state his main result (due to Ajit Sanzgiri) is that "Groups of homotopy spheres are homotopy groups of spheres".

Milnor received the Fields' Medal in 1962. In addition, the work of a number of later Fields' medallists such as Donaldson, Thurston, Mori and Voevodsky can be seen as having roots in the work of Milnor.

[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!