Revision 1: Inquisitive Mathematical Thinking

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.”.

Popular-Lonely primes understood

While reading standup mathematician Matt Parker‘s book Things to Make and do in Fourth Dimension, I found answer (on pp. 146) to the question I raised 7 months ago.

When the grid happens to be a multiple of 6 wide, suddenly all primes snap into dead-straight lines. All primes (except 2 and 3) are one more or less than a multiple of 6. (© Matt Parker, 2014)

He also proves the following surprising theorem:

The square of every prime number greater than 3 is one more than a multiple of 24.

Let be an odd prime not equal to 3. Now we subtract one from the square of this prime number. Therefore, we wish to prove that is a multiple of 24.

Note that, is a product of two even numbers. In particular, one of these two even numbers must be a multiple of 4, as they are consecutive even numbers and every other even number is divisible by 4. Hence we conclude that is divisible by 8. 

Observe that exactly one of three consecutive numbers, must be divisible by 3. Since is an odd prime different from 3, one of or must be divisible by 3. Hence we conclude that is divisible by 3.

Combining both the conclusions made above, we complete proof of our statement (since 2 and 3 are coprime).

Edit[19 April 2017]: Today I discovered that this theorem is exercise 68 in “The USSR Olympiad Problem Book“.

Special Numbers: update

This post is a continuation of my earlier post: Special Numbers

Four (4)

This is the only euclidean space with properties different from other n-dimensional euclidean spaces. For example, there are smooth 4-manifolds which are homeomorphic but not diffeomorphic.  Put differently, for any dimension except four there is only one differentiable structure on the space underlying the Euclidean space of that dimension. For a discussion in this direction see this article by Liviu Nicolaescu. For other special properties of 4-dimesnions read Wikipedia article on 4-manifold.
Thanks to Dr. Ritwik Mukherjee for explaining this fact about four-space.

Hyperbolic Plane Example

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 :

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.