Friday, 16 November 2012

Something about nothing and everything

zinfty

The word finite in English is an adjective that means limited or having bounds or one that has an end. In mathematics, this adjective is used often, viz., finite sets, finite dimension etc. Further, to refer to things that are not finite we have, of course, an adjective which is the antonym to finite, called infinite. This adjective means limitless or boundless or endless. In Sanskrit and Tamil, the equivalent word for infinite is ananta. The word ant means end and anant means that which has no end. Both finite and infinite is a measure of size (large or small). Infinite is larger than all finite things.

The word infinity is a noun which refers to the state of being infinite. The perceptions of any organism on this planet, in normal circumstances, have only finite capabilities. For instance, we know today that human eye can respond to wavelengths between 390–750 nm, the audible range for human ear is between 20Hz–20kHz. Humans can distinguish six types of taste. Similarly, human smell and feel are all limited. This limitation is also true with our logical/rational thinking, usually considered the sixth sense. This attribute is, naturally, inherited to human inventions as well. For instance, computers cannot deal infinity. The number may be as large as 10500 today, but still finite! Telescopes can see farther in the sky today, still finite! Thus, in my opinion, understanding infinity may not be possible using logical/rational mind. Let me explain myself.

Consider the two words nothing and everything. It is clear that they are antonyms to each other. Thus, we need to define one of these words and the definition of the other follows naturally as its negation. But you see there is a philosophical problem in defining either of them consistently. Suppose, we say, nothing is that which is not a thing, then the fact that nothing has a definition means it is something and hence cannot be nothing. If, on the contrary, we try to define everything as that which includes or considers all the things, then does everything include itself? Also, by definition, it should include its antonym nothing. Is this the Russell’s paradox? Hold! Don’t throw things at me. I’m not yet done!

In mathematics, we do know that things that are so natural to us begins to fail in an infinite set-up. For instance, we know 1+2+3 = (1+2)+3 = 1+(2+3) = (1+3)+2 =6. Simply put, in whatever order you sum, you will get the same answer. This is called the distributive and commutative property of natural (or real) numbers. But this would fail in an infinite set-up. If not, we should have

0 =0 + 0 + 0 + …    (Infinite sum)
 =(1 −1) + (1−1) + (1−1) + …
 =1 +( −1 +1) +  (−1 +1) + (−1 + 1) + …    (rearranging the terms)
 =1 + 0 + 0 + 0 + …
 =1.

We have proved 0=1! Or have we? If you want to see more such counter-intuitive stuffs, do a course (or read a book) on basic real analysis, especially limits, sequences and series. Anyway, back to our argument of 0=1. You see the moment we carry forward notions from finiteness to infinity, we get absurd(?) things like 0=1. So, what went wrong? We had conveniently assumed, in first equality, 0 is same as sum of 0 infinite times. Was that right? May be not. If nothing has to yield nothing, however infinite, then modern Big Bang theory is a contradiction. Ancient Indian texts claim that everything comes from nothing and goes back to nothing. That is why the Indian symbol of nothing is a circle (0). Lord Shiva is considered to have no aadi (beginning) and no antam (end) and a mythological story depicts this where Lord Brahma and Vishnu go in search of His aadi and antam, respectively. One should view mythological stories in ancient Indian texts as fables for adults!1 One meaning of the word Shi Va is “that which is not or does not exist”, referring to nullity and at the same time by saying it has no beginning and no end they are referring to infinity. Thus, they seem to imply the duality of nullity and infinity. In mathematics, this duality is exhibited by the function f(x) = 1/x. Also, infinity is omnipresent. Mathematically, we know that the cardinality of [0,1] is same as the real line ℝ but [0,1] is properly contained in ℝ. Consider the analogy of ℝ being universe. I know, I know, I’m talking absurd. The power set of ℝ is bigger than ℝ. Just take the analogy. I’m just illustrating a property of infinity. You can work with the power set of ℝ or its power set or its. The property will remain true.

So, back to our analogy. If ℝ denotes the universe and every unit (or any length, just an example) interval [a, a+1] represents an organism in this universe. Then, every organism has a universe within itself. In other words, exploring universe is same as exploring within. Though you are part of the universe, you have the universe within you; something like [0,1] and ℝ. This is property of infinity. The set of even (or odd) integers is a proper subset of all integers but have the same cardinality. (I’m going to be torn apart for the contents of this paragraph!). On the other hand, it is the beauty of mathematics that discretises infinity using cardinal numbers.

Let’s ponder further on our argument that 0=1, assuming nothing yields nothing, however infinite. The second equality is not a problem, just writing each zero as sum of two terms. The third equality is rearrangement. Mathematically, this is where the main problem lies. If the rearrangement was true, then we could have also rearranged the term after second equality as

(1+1+ …) − (1 + 1 +  …) = ∞ − ∞,

difference of divergent series and that is not zero (why?). Of course, mathematics fixes such issues by studying absolutely convergent series, uniform convergence of functions, dominated convergence result of Lebesgue etc. But, in my opinion, these are just isolating particular cases where infinity behaves like finite. Axiom of Choice is nothing but imposing one property of finiteness to infinity. One of its many consequence, the Banach-Tarski paradox is again due to infinity. We conveniently call some guys, who don’t fit in our understanding, as bad guys (non-measurable sets) to come up with a consistent theory that fits with what we see as reality. Note that I said “what we see as reality”.

Yeah, yeah! I’m going to conclude. My assertion is that our attempt to study infinity using logic is like trying to study infinity using finiteness and this may be an impossible situation. So, what is the right way to handle infinity? Of course, I don’t know. Else I would be in the hall of Nobel laureates. But identifying wrong paths is the first step in right direction.


1
The ancient science took the form of religion for political reasons like, invasions and let us not digress there now.