Mathematical Infinity and Philosophical Infinity

[Guest guest]

Namaste

 

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 19th 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 revcolutionary 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!

 

 

PraN Ams to all advaitins

profvk

 

 

 

 

=====

Prof. V. Krishnamurthy

My website on Science and Spirituality is http://www.geocities.com/profvk/

You can access my book on Gems from the Ocean of Hindu Thought Vision and

Practice, and my father R. Visvanatha Sastri's manuscripts from the site.

Also see the webpages on Paramacharya's Soundaryalahari :

http://www.geocities.com/profvk/gohitvip/DPDS.html

[Guest guest]

thank you professorji !

 

you know i hold you in very high esteem so whatever you say i acceot

it as the 'gospel'.i was never good at mathematics ( specially

Calculus - i was good at trignometry, geometry and algebra). Some of

this discussinon here goes right over my head that is because i

am 'dumb' in the matter of numbers!

 

however , how do you react to this ?

 

 

"The emergence of numbers is traced from purnam. Hence, the dependence

of the Darsana on purnam, "Yadvai purnalambam" (A-2). Nrisimha has

given an interesting example to show how! It is like the sprouting

power lying hidden in the seed. Under favourable conditions the seed

sprouts and grows into a big tree. Likewise purnam lies hidden in

Samkhya and gives birth to an infinite number (phenomenal universe),

Taddhi mulatatvam samkhyayah (A-4). Like Brahma it has existence but

invisible(M-5). It was the primal principle, which gave rise to

numbers, Ankas 1-9 and entered those numbers like the invisible Sat

(existence) and created a whole world of numbers like 10, 20, ……..~ .

It is in accordance with Veda – tat srishtva tadevanupravisat;

tadanupravisya sat cha tyat cha abhavat ( sat = existence, tyat =

illusory world)."

 

 

please explain in your inimitable loving fashion

 

namashkarams

 

,

[Guest guest]

Namaste Prof. Krishnamurthyji.

 

Thanks for your two wonderful posts on algebra and infinity.

 

I brought in mathematics in my lead post just to point out how

easily very intelligent mathematicians get carried away in wrong

directions. I never anticipated a wild and headless discussion.

 

Like Kenji, I am also concerned and, as initiator of the discussion,

request everyone to kindly put an end to this unfortunate digression

and to continue the discussion with the respect due to Infinity as

one without a second beyond the realm of mathematical jugglery.

 

PraNAms.

 

Madathil Nair

_____________________

 

advaitin, "V. Krishnamurthy" <profvk>

wrote:

.........> 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. ...........

> 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. .......................

>