pUrNamadah pUrNamidam .......... Infinity: this is more mathematical

[Guest guest]

Namaste all,

 

There have been some discussions on the mathematical view of Brahman,

and it quite impresses me to know that many here are mathematics guys

like professorji. I thought, without advising anyone or without

trying to appear too cheeky, if I make a comment, the group would not

mind it..... I hope....

 

I know mathematics is slightly disliked by a small section of people,

but I beleive they have a misplaced dislike for it. Anyway that is

besides the point. But I do want to make sure that it is not against

the spirit of the group to discuss it. Let us not think that such

intellectual speculations on inifinty will enlighten one. The

knowledge required for enlightenment must come from beyond the

intellect, for Brahman himself is beyond the intellect.

 

First I shall note that algebra was (probably) independently

developed both in ancient India and in some parts of Persia. However,

its history is not as important. I do beleive that algebra was surely

a part of ancient Indian studies, since I know quite a bit of Vedic

Mathematics and do constantly use it.

 

The deductions on infinity based on bIja gaNita, are not based on

algebra. One has to understand bIja gaNita more to say that. Bhaskara

never attempted to quantify infinity as an algebraic entity. That is

because inifinity is not an algebraic element at all. I hope I am not

offending anyone. This is of course, based on my own study of ancient

Indian mathematics.

 

But let us go beyond the Indian mathematics scenario and understand

what mathematics in general has to say about infinity.

>From the understanding of real analysis, infinity represents the

entire real line. Here, the concept of infinity has more to do with

boundedness than with a concept of division as has been incorrectly

percieved. Of course there is the concept of countability also. I

shall adress all these issues one by one.

 

THE DIVISION BY ZERO

=====================

 

The concept of infinity is different from that of a number being

divided by zero. The division itself is not a valid operation under

such circumstances. How could it be infinity? The statement should be

revised:

 

1/0 is undefined (in mathematics atleast)

1/r, where r is a very small quantity, is not infinity, but

1/r, where r is a very small quantity, tends to infinity, in the

limit! There is a difference. This concept would be clear to those

knowing calculus or set theory or the concept of limit. It shows that

in the case of division, by a very small number, infinity is still

understandable only in the limit! This means that 1/r is not equal to

infinity. Infinity cannot be equated to any algebraic quantity.

 

This understanding of infinity will be useful, when I point out the

way in which mathematics agrees with the pUrNamadah verse.

 

REAL ANALYSIS ON INFINTY

==========================

 

The way infinity has been defined is:

 

'If a number M belongs to the real number set such that for every

real number m, M is greater than m, then M tends to infinity.'

 

The point to be noted is that even such a very large number, which is

larger than all other numbers is not said to be equal to infinity, it

is said to only tend to infinity.

 

The problem is, the concept of infinity is rather complex and is not

very clear in mathematics, since our intellect has limited capacity

to follow it. The bodha (according to Buddha) or jnah (according to

Sankara), which are both indicative of Brahman are again names used

for this infinite (anantha or pUrNa). Only this bodha or jnah is

capable of knowing the infinite.

 

pUrNasya pUrNamAdaya pUrNamevAvashishyate

 

Real analysis agrees here too.

 

infinity + infinity = infinity

 

What is infinity - infinity?

 

When the question of subtraction is involved, the amateur algebra

student, will make the error and say, it will result in zero. But

that is not the case. This cannot be understood from the standpoint

of an upper bound. One needs to know the point of countability of

sets for this.

 

Consider the entire real line, how many elements are there? Infinite

Consider the interval between 0 and 1, how many elements are there?

Again infinite? Which infinite is bigger? One may be tempted to say

that the former is bigger, but the right view, when adopted on the

countability of sets, shows that there are the same number of

elements between 0 and 1 as there are on the entire real number line.

That's amazing. That's how complex the concept of infinity is. If

someone does not understand how the interval has the same number of

elements, he may ask me to explain. If the moderators feel it is an

unnecessary digression, I shall send it as a personal post to all

those requesting an explanation.

 

Now let me ask you a question, based on the understanding that the

number of elements in an interval on the number line is equal to

those on the entire number line. Let me remove a small interval from

the number line, and join the cut off portions (eliminate the gap

thus created). Remember the entire real set had the same number of

elements as the interval itself. Now that the interval is removed, is

the remaining set of real numbers, void? No. There are still infinite

numbers there.

 

Hence from here, we see that infinity - inifinity is also infinity.

Hence:

 

infinity - infinity = infinity

 

Also we know that

 

infinity + any other number = infinity

 

OTHER OPERATIONS ON INFINTY

==============================

 

1. Is infinty * 0 = 0?

 

The operation * when defined on algebraic quantities says that any

algebraic number x multiplied by zero gives zero. But infinity is not

an algebraic quantity. Hence the result cannot be zero. Then what is

it? It has no result uniquely expressible in mathematical terms.

 

2. Is infinity / infinity = 1?

Yes. But the form, infinity/infinity in the case of the limit is

different. There, neither the numerator nor the denominator are equal

to infinity. They are just tending towards infinity. When infinity

itself is to be divided by infinity, we will get only unity.

 

This may be funny, since the process of division seen as successive

subtraction, by laymen will lead to thinking that infinity / infinity

= infinity. But division is not sucessive subtraction, it is an

operation of partitioning. I don't know if I have convinced you of

the last point. If not, please let me know. I'll try again.

 

Incidentally, the last point is again important for the Advaitin

group, since it tells that Atman and Brahman, both being infinite are

one and the same. But Atman cannot be subtracted from Brahman, if we

do so, we will still be left with Brahman.

 

Satyameva Jayate Naanrtam

[Guest guest]

Namaste!

 

Thank you, Shri balaji, for a brilliant explanation of infinity. I

was reminded of my maths professor in college who constantly told us

that infinity is not an algebraic quantity. If anyone in class made

the mistake of using the symbol for infinity in an expression while

answering a test, he was sure to fail the test!

 

The idea of 'purnatvam' has become a little clearer to me now. Thank

you again.

 

Hari Om.

Neelakantan