**MATHEMATICAL
& PHILOSOPHICAL INFINITIES**

**(Copy of
Post #22191 dated April 13, 2004 on the “advaitin” list.)**

I apologise first for not intruding earlier in the discussion of ‘Infinity’ that is going on. Too many posts have come out on infinity. And some of the statements are not mathematically acceptable. As one who has professed mathematics all my life, I would be betraying my Queen of Mathematics if I sit here without correcting some misinterpretations. Hence this post. Pardon me if it goes outside of advaita. You asked for it!

The fundamental misinterpretation with which mathematics teachers are not unfamiliar and which has crept into these discussions is: Infinity being talked about as if it were a number like 1, 2, 3, ...

The concept of a number itself is
not simple. Mankind had to wait for
Georg Cantor of the 19^{th} century to arrive at a precise concept of a
number. To start with one has to realise that the familiar number, say, ‘five’
is not the symbol ‘5’ – which may be written differently in different cultures
– but it is the commonality that exists between all sets which can be put in
one-to-one correspondence with the set of fingers on a normal human hand. The
concept of ‘set’ and of ‘one-to-one correspondence of sets’ were introduced by
Cantor for the first time in the world of Mathematics. When applied to sets which are finite, the
concept of ‘one-to-one correspondence may appear to be a triviality. But when
we have to talk about the counting of infinite sets like the set {1, 2, 3,
.....} (or even ‘larger’ infinite sets)
we would begin to comprehend the necessity for the concept of
‘one-to-one correspondence’. Ancient Asian texts do talk about very large
numbers (Brahma’s age and so on) but when it comes to ‘infinite numbers’ they
get into the philosophical ‘pUrnaM’, but not the mathematical ‘Infinity’.

And Cantor created history by proving (!) – Yes, his mathematical proof created the first modern revolution in mathematics – that there are *different* kinds of infinite sets.

An important digression: In the first course on Calculus you are taught about functions which ‘tend to infinity’. This is only a way of saying something which takes very precise formulations in mathematics. It does not mean that infinity is a number. Infinity is never an ordinary number in all of mathematics. That is why “Infinity divided by infinity” is a misnomer in mathematics. The infinity (and infinities) that Cantor talked about are called ‘Cardinal numbers’; their algebra is significantly different from the ordinary algebra applicable to numbers. To tread that path one has to come via Cantor’s path.

Cantor’s first revolutionary statement and proof was about ‘the set of natural numbers {1,2,3, ...}’ and ‘the set of all positive fractions(=ratios) of natural numbers’. He created a one-to-one correspondence between the two sets and thus proved that the two sets are ‘equally infinite’. NOTE: I am using some loose descriptive words instead of the correct mathematical expressions.

And the next revolutionary result of his was that the infinity represented by the set of natural numbers and the infinity represented by all positive numbers including all fractions and all decimals are two distinct ‘infinities’. This made him define a cardinal number and he went to town by creating a whole host of cardinal numbers – each of which was a different order of infinities.

Oh, there is a lot more. But I will not tire your patience. Those of you who want to learn more about these infinite cardinal numbers may just type ‘Countable and Uncountable sets’ in Google and I am sure you will be led on to elaborate presentations of the topic.

Now comes the Philosophical Infinity (pUrnaM). PUrnaM is the Absolute. It is undefinable. The mathematical Infinity (of Cantor) on the other hand can be precisely formulated in words. The philosophical infinity is ‘anirvacanIya’. Words ‘return’ from It. “yato vAco nivartante”. To equate the mathematical infinity and the philosophical infinity is to commit a sacrilege.

And then, to boot, there is a third type of infinity, namely, physical infinity. Rudy Rucker in her book ‘Infinity and the Mind’ (1982) talks about eight possibilities of existence of these three types of infinities – mathematical, physical and philosophical. There are scientists who hold all three exist, and there are scientists who hold that none of these exist. In between there are the other possibilities, making a total of eight. A footnote in her book suggests the names of the eight representative scientists for these eight possibilities: Abraham Robinson, Plato, Thomas Aquinas, L.E.J. Brewer, David Hilbert, Bertrand Russell, Kurt Godel and Georg Cantor.

I think I should stop here. There are wonderful books that can be seen on the subject:

1.Courant and Robbins: What is Mathematic s? 1941

2.E.T. Bell: Mathematics, Queen and Servant of Science.

Of course I can also refer you to my own book “Culture, Excitement and Relevance of Mathematics” (1990) where Chapter III, Section 1. talks about Cantor and his cardinals. The book is available in Amazon.Com – maybe only used copies!

Copyright © V. Krishnamurthy January 2007