Monthly Archives: November 2016

Introspection

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I have written many philosophical blog posts motivated by the idea of existence of reason for everything.  So far I have written (and revised) my point of view regarding topics like Being Alive, Loving MathematicsBecoming Mathematician and Dreaming Big.

A more fundamental and more disturbing question is:

Since each and everything is made up of atoms, what is the borderline between a living organism and non-living object.

Clearly, mathematics will fail to answer this question since there are no absolute quantifiers involved. But still we can try deducing an answer from logical arguments. Following video by Kurzgesagt – In a Nutshell illustrates my question:

We generally describe living organism as something capable of reproduction, growth and consciousness. So, can we convert anything into living organism by somehow adding artificial intelligence to it? Is internet itself a living organism?

Once you call yourself a living organism, the immediate question is about the purpose of your existence (since we believe that there is reason for everything). So we can use this as a quantifier to classify something as living and non-living. Many people have tried (and failed) to answer this question. I came across a possible answer for this question in the film Kubo and the two strings:

We live to write a story and then become immortal in memories of others in form of our stories.

kubo-two-strings-quote

Kubo and the two strings (© 2016, Focus Features)

I really liked this point of view. Being alive is all about being able to create memories. But this view point is very much human centred since we don’t know how other organisms (like other animals, plants, cells, organelles…) interact. Moreover, non-living objects also have stories associated with them (like monuments, paintings,…). So this view point also fails to capture the central idea for classification of something as living or non-living.

I will be happy to know your viewpoint of being able to classify something as living or non-living.

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Not all numbers are computable

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When we hear the word number, symbols like 1,\sqrt{2},¼, π (area enclosed by a unit circle), ι (symbol for \sqrt{-1}), ε (infinitesimal),  ω (ordinal infinity), ℵ (cardinal infinity), …. appear in our mind.  But not all numbers are  computable:

A computable number [is] one for which there is a Turing machine which, given n on its initial tape, terminates with the nth digit of that number [encoded on its tape].

In other words, a real number is called computable if there is an algorithm which, given n, returns the first n digits of the number. This is equivalent to the existence of a program that enumerates the digits of the real number. For example, π is a computable number (why? see here).

Using Cantor’s diagonal argument on a list of all computable numbers, we get a non-computable number (here is the discussion). For example, a sum of series of real numbers called Chaitin’s constant, denoted by Ω, is a non-computable number (why? see here).

Fun fact: We don’t know whether π is a normal number or not (though we want it to be a normal number), but Ω is known to be a normal number (just like the Mahler’s Number discussed here).

Celebrity Mathematicians

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In my opinion, currency notes are one of the biggest motivation to learn arithmetical operations (like addition, multiplication,…). In fact, most of our elementary school problems are about buying a particular quantity of something.

Historically, there had been currencies notes featuring great mathematicians like Carl Friedrich Gauss, Leonard Euler and Rene Descartes. But, today there are no currencies featuring mathematicians. The database of currency notes featuring mathematicians is available here: http://web.olivet.edu/~hathaway/math_money.html

Since honouring people by featuring them on currency notes is politically challenging, government rather issues special postage stamps. The database of stamps featuring mathematicians is available here:  http://jeff560.tripod.com/stamps.html

Apart from illustrating various mathematical concepts (like graphs, metric system, binomial theorem… ) on stamps, India Post has issued stamps to honour mathematicians like Damodar Dharmananda Kosambi , Srinivasa Ramanujan Iyengar and Bertrand Russell.

 

Real Numbers

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Few days ago I found something very interesting on 9gag:

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When I first read this post, I was amazed because I never thought about this obvious fact!!

There are lots of interesting comments, but here is a proof from the comments:

…. Infinite x zero (as a limit) is indefinite. But infinite x zero (as a number) is zero. So lim( 0 x exp (x²) ) = 0 while lim ( f(X) x exp(X) ) with f(X)->0 is indefinite …

Though the statement made in the post is very vague and can lead to different opinions, like what about doing the product with surreal numbers, but we can safely avoid this by considering the product of real numbers only.

Now an immediate question should be (since every positive real number has a negative counterpart):

Is the sum of all real numbers zero?

In my opinion the answer should be “no”. As of now I don’t have a concrete proof but the intuition is:

Sum of a convergent series is the limit of partial sums, and for real numbers due to lack of starting point we can’t define a partial  sum. Hence we can’t compute the limit of this sum and the sum of series of real numbers doesn’t exist.

Moreover, since the sum of all “positive” real numbers is not a finite value (i.e. the series of positive real numbers is divergent) we conclude that we can’t rearrange the terms in series of “all” real numbers (Riemann Rearrangement Theorem). Thus the sum of real numbers can only be conditionally convergent. So, my above argument should work. Please let me know if you find a flaw in these reasonings.

Also I found following interesting answer on Quora:

The real numbers are uncountably infinite, and the standard notions of summation are only defined for countably many terms.

Note: Since we are dealing with infinite product and sum, we can’t argue using algebra of real numbers (like commutativity etc.).