Tag Archives: mathematics

Evolution of Language

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We know that statistics (which is different from mathematics) plays an important role in various other sciences (mathematics is not a science, it’s an art). But still I would like to discuss one very interesting application to linguistics. Consider the following two excerpts from an article by Bob Holmes:

1. ….The researchers were able to mathematically predict the likely “mutation rate” for each word, based on its frequency. The most frequently used words, they predict, are likely to remain stable for over 10,000 years, making these cultural artifacts, or “memes”, more stable than some genes…..

2. ….The most frequently used verbs (such as “be”, “have”, “come”, “go” and “take”) remained irregular. The less often a verb is used, the more likely it was to have been regularised. Of the rarest verbs in their list, including “bide”, “delve”, “hew”, “snip” and “wreak”, 91% have regularised over the past 1200 years…….

The first paragraph refers to  the work done by evolutionary biologist Mark Pagel and his colleagues at the University of Reading, UK. Also, “mathematically predicted” refers to the results of the statistical model analysing the frequency of use of words used to express 200 different meanings in 87 different languages. They found the more frequently the meaning is used in speech, the less change in the words used to express it.

The second paragraph refers to the work done by Erez Lieberman, Jean-Baptiste Michel and others at Harvard University, USA.  All people in this group have mathematical training.

I found this article interesting since I never expected biologists and mathematicians spending time on understanding evolution of language and publishing the findings in Nature journal. But this reminds me of the frequency analysis technique used in cryptanalysis:

Building Mathematics

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Let’s talk about the work of a mathematician. When I was young (before highschool), I used to believe that anyone capable of using mathematics is a mathematician. The reason behind this was that being a mathematician was not a job for people like Brahmagupta, Aryabhatta, Fermat, Ramanujan (the names I knew when I was young). So by that definition, even a shopkeeper was a mathematician. And hence I had no interest in becoming a mathematician.

Then, during highschool, I came to know about the mathematics olympiad and was fascinated by the “easy to state but difficult to solve” problems from geometry, combinatorics, arithmetic and algebra (thanks to AMTIVipul Naik and Sai Krishna Deep) . I practiced many problems in hope to appear for the exam once in my life. But that day never came (due to bad education system of my state) and I switched to physics, just because there was lot of hype about how interesting our nature is (thanks to Walter Lewin). 

In senior school I realised that I can’t do physics, I simply don’t like the thought process behind physics (thanks to Feynman). And luckily, around the same time, came to know what mathematicians do (thanks to Uncle Paul). Mathematicians “create new maths”. They may contribute according to their capabilities, but no contribution is negligible. There are two kinds of mathematicians, one who define new objects (I call them problem creators) and others who simplify the existing theories by adding details (I call them problem solvers). You may wonder that while solving a problem one may create bigger maths problems, and vice versa, but I am talking about the general ideologies. What I am trying to express, is similar to what people want to say by telling that logic is a small branch of mathematics (whereas I love maths just for its logical arguments).

A few months before I had to join college (in 2014), I decided to become a mathematician. Hence I joined a research institute (clearly not the best one in my country, but my concern was just to be able to learn as much maths as possible).  Now I am learning lots of advanced (still old) maths (thanks to Sagar SrivastavaJyotiraditya Singh and my teachers) and trying to make a place for myself, to be able to call myself a mathematician some day.

I find all this very funny. When I was young, I used to think that anyone could become a mathematician and there was nothing special about it. But now I everyday have to prove myself to others so that they give me a chance to become a mathematician. Clearly, I am not a genius like all the people I named above (or even close to them) but I still want to create some new maths either in form of a solution to a problem or foundations of new theory and call myself a mathematician. I don’t want it to end up like my maths olympiad dream.

Teaching Mathematics

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One of the most challenging and rewarding thing associated with being a math enthusiast (a.k.a. mathematician) is an opportunity to share your knowledge about the not so obvious truths of mathematics. A couple of years ago, I tried to communicate that feeling through an article for high school students.

When I joined college, I tried to teach mathematics to some kids from financially not-so strong family. Since they had no exposure to mathematics, I had to start with  concepts like addition and multiplication of numbers. My experience can be summarized as the following stand-up comedy performance by Naveen Richard:

After trying for about a couple of months to teach elementary mathematics, I gave up and now I discuss mathematics only above the high school level. Last week I delivered a lecture discussing the proof of Poncelet’s Closure Theorem:

Whenever a polygon is inscribed in one conic section and circumscribes another one, the polygon must be part of an infinite family of polygons that are all inscribed in and circumscribe the same two conics.

I had spent sufficient time preparing the lecture, and believed that I was aware of all possible consequences of this theorem. But, almost half way through the lecture one person (Haresh) from the audience of 10 people, pointed out following fascinating consequence of the theorem:

If an n-sided polygon is inscribed in one conic section and circumscribed by the other one, then it must be a convex polygon and no other m-sided polygon (with m≠n) can be inscribed and circumscribed by this pair of conic sections.

This kind of insights by audience motivates me to discuss mathematics with others!

Revision 1: Inquisitive Mathematical Thinking

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In this post I wish to expand my understanding about, “asking Why?“.

In 1930, David Hilbert gave radio address lecture. I want to discuss following paragraph from that lecture (when translated to English):

With astonishing sharpness, the great mathematician POINCARÉ once attacked TOLSTOY, who had suggested that pursuing “science for science’s sake” is foolish. The achievements of industry, for example, would never have seen the light of day had the practical-minded existed alone and had not these advances been pursued by disinterested fools.

Science exists because we (human beings) want to find reason for everything happening around us (like how air molecules interact, which bacteria is harmful…) . We claim that this will enrich our understanding of the nature thus enabling us to make rational decisions (like when should I invest my money in stock market, from how much hight I can jump without hurting myself…).

Let me illustrate the point I want to make: Mathematicians make observations about real/abstract objects (shape of universe/klein bottle) and try to explain them using logical arguments based on some accepted truths (axioms/postulates). But today we have “science” for almost every academic discipline possible. Therefore, we (human beings) have become so much obsessed with finding reasons for everything that we even want to know why the things happened a moment ago so that we are able to predict what will happen in a moment from now. So the question is:

Should there be a reason for everything?

Can’t some thing just be happening around us for no reason. Why we try to model everything using psedo-randomness and try to extract a meaning from it? In case you are thinking that probability helps us understanding purely random events, you are wrong. We assume events to be purely random, we are never sure of their randomness and based on this assumption we determine chances of that event to happen which infact tells nothing about future (like an event with 85% chances of happening may not happen in next trial).

In same spirit, I can ask: “Should there be reason for you being victim of a terrorist attack?” We can surely track down a chain of past events (and even the bio-chemical pathways) leading to the attack and you being a victim of it.

Why we try to give “luck” as reason for some events? Is this our way of acknowledging randomness or our inability to find reason?

Moreover, David Hilbert ends his lecture with following slogan (in German):

Wir müssen wissen, Wir werden wissen.

which  when translated to English means: “We must know, we will know.”.

Hyperbolic Plane Example

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Few months ago I gave a lecture on Non-euclidean geometry and it was a bit difficult for me to give audience an example of hyperbolic surface from their day-to-day life. While reading Donal O’ Shea’s book on Poincaré Conjecture I came across following interesting example on pp. 97 :

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Negatively curved cloth will drape a woman’s side (© Donal O’ Shea, 2007)

Estrogen causes fat to be stored in the buttocks, thighs, and hips in women. Thus females generally have relatively narrow waists and large buttocks, and this along with wide hips make for a wider hip section and a lower waist-hip ratio compared to men. The saddle-shaped area on a woman’s side above her hip has negative curvature.

One can imagine cloth (it is flexible but does not stretch, hence an isometry) that would drape it perfectly. Here the region inside a circle of given radius contains more material than the same circle on the plane, and to make the cloth the tailor might start with a flat piece of fabric, make a cut as if he/she were going to make a dart, but instead of stitching the cut edges together, insert an extra piece of fabric or a gusset. Negatively curved cloth would have lots of folds if one tried to lay it flat in  dresser.

If one tries to extend a cloth with constant positive curvature (like a cap), in all directions, it would close up, making a sphere. On the other hand, if one imagines extending a cloth with constant negative curvature in all directions, one gets a surface called hyperbolic plane.

Mathematical Relations

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In this post I will share my perception of relation of mathematics with other academic disciplines. All this is based on my very limited knowledge of various disciplines.

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Shape doesn’t signify anything.

Mathematics deals with study of properties of numbers (or the symbols representing them) and geometric objects (not in classical sense, it can mean manifolds also). In my opinion, there is no partition of mathematics into “applied” or “pure”, but intersections with other subjects. The term applied Mathematics doesn’t make any sense to me. Mathematics is somehow applicable in various places. For me, mathematics is what people call “pure” mathematics (what about “impure” Mathematics??).  Also now I agree with the vastly established belief that art and mathematics are similar, since both involve abstract ideas motivated but physical situations (at some point).

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Truth Lies Deception and Coverups – Democracy Under Fire (Source: http://goo.gl/yUHi93)

All experimental sciences (physics, chemistry, biology, economics) are based on statistics. Since statistics is a young discipline (only a couple of centuries old) many times we get wrong interpretation of results. As far as real life is concerned, study of statistics gives us a powerful tool for predicting future and Probability Theory acts as the connecting link between statistics and mathematics. Understanding of statistics affects us on daily basis since (effective) government policies are framed keeping statistical analysis in mind. Unfortunately, most of universities don’t have separate department for statistics.

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P vs NP Problem in Relationships (http://ctp200.com/comic/6; CC BY-NC 4.0)

Study of algorithms is one of the most important aspect of computer science (I am not talking about software industry…). What surprises me is that Euclid’s division algorithm is  one of the most efficient division algorithm even for computers! The neglected subject of Logic, which is supposed to be foundations of mathematics, flourishes in computer science. P vs NP is another “millennium open problem“.

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Convincing (http://xkcd.com/833/ ; CC BY-NC 2.5)

For me, Economics like Statistics is full of imperfections due to real life complications (so many dependencies to account for). Game Theory appears to be the connecting link between mathematics and economics.

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We all know that the needs of physicists are responsible for development of calculus and study of differential equations. On the other hand, theoretical physics (quantum mechanics, string theory) depends heavily on the developments in algebra.

Mathematics Today…

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When we talk about doing mathematics, what comes to  our mind is Blackboard-chalk, notebook-pen and books. No doubt that these are and will remain one one of the most important instruments leading to elegant mathematical discoveries.  But, the evolution of technology we use has also affected the way we do, learn and share mathematics. In ancient time, mathematics was shared in form of books and letters. Then in 17th Century people started publishing academic journals periodically, which has today become one the most profitable business (like pharmaceuticals).  In 1960s computer algebra systems were invented (called MATHLAB). Then in 1970s books were digitized and today we have dedicated ebook readers.  Another major challenge  of publishing mathematical knowledge was to be able to typeset weird symbols, and this problem was fully solved using computers in 1978 by Donald Knuth. (to know more about this transition read this discussion).

Now it’s 21st century and the shape of sharing mathematical knowledge has changed significantly in past decade.  To begin with, in 2003 Poincaré conjecture’s solution was not published in any journal but was rather posted on arXiv. Today we have people on social networking sites like Facebook, Twitter, Google+, Tumblr, Weblogs...  who let you know the results just as they are being cooked up. For example, Live-tweet of Babai’s first Graph Isomorphism talk, in this talk one of the most interesting theorem of 2015 was proved. Many big shots announce their big results directly on their Weblogs, for example Terence Tao announced his proof of Erdős Discrepancy Problem on his blog. Today we can have interactive textbooks (like this), articles (like this) and assignments (like this) with advent of MathJax, SageMath

So far I have been concerned about “print” mathematics, but with advent of cheap internet, whole new methods of mathematical ideas sharing have come into picture. Today almost every reputed research organization maintain video lecture archives (IAS, CIRM,  IHÉSIHPInstitut Fourier, MatScience). Apart from mainstream mathematics, popularization of mathematics has become much more interesting today. We have lots of interesting mathematics popularization channels on YouTube like ViHartNumberphile, Mathologer, 3Blue1Brown, The Global Math Project,… and SoundCloud like BBC Radio 4: More or Less, ACMEScience ,… For a big-list of online mathematics videos see this and for big-list of mathematics podcasts see this.

Before this internet era, there were similar mathematics popularization attempts. Like my favourite: “Donald Duck in Mathmagic Land” (1959) [updated the link on 28 Dec’18]

But I’m not aware of existence of mathematical radio programs back then. So, if you know about such radio programs,o please let me know about them as comments below.