The Challenges of Mathematics

The Challenges of Mathematics

The progress of human civilization largely depends on the advancement of science and technology which advance on the wheels of mathematics.

Ever since the dawn of civilization, mathematics has played a vital role in the transformation of our society. But what propelled mathematics to flourish?

The obvious answer is challenging problems fed by all branches of sciences and social sciences but mainly by physics, chemistry, biology and economics. Mathematics of all ages took these problems as challenges and endeavored to solve them as a result of which new theories evolved, new techniques were developed and new concepts were coined.

Thus the solutions to these problems enriched mathematics, often science also. Some of these problems were revolutionary also in the sense that they changed the mathematical paradigm, extending the horizons of mathematics and finally giving precise interpretations of the physical and social phenomena.

Many of such problems are still unsolved and attempts to solve them have given us new methods and often new theories. Mathematicians took several years to solve them; some problem even took several centuries.

Euclid

Ever since the dawn of civilization, mathematics has played a vital role in the transformation of our society. But what propelled mathematics to flourish? The obvious answer is challenging problems fed by all branches of sciences and social sciences but mainly by physics, chemistry, biology and economics. Mathematics of all ages took these problems as challenges and endeavored to solve them as a result of which new theories evolved, new techniques were developed and new concepts were coined.

The challenges that changed the world:

Some of the outstanding problems whose solutions brought about a phenomenal change in mathematics are the following:

(a) In Euclid’s Geometry the following three problems posed great difficulty to mathematicians of several centuries:

(i) Trisection of an arbitrary angle with ruler and compass only;

(ii) Drawing a square equal in area to a circle using ruler and compass;(iii) Constructing a cube whose volume is double the volume of another cube.

These problems evolved in early first millennium A.D. and took nearly ten centuries to find their solution till Galois gave a formal solution in the negative.

(b) In Number Theory the following two problems perturbed almost all mathematicians of several centuries:

Goldbach

(i) Goldbach’s Conjecture: Every even natural number greater than 2 is expressible as the sum of two prime numbers. Mathematician Goldbach posed this problem in the year 1742 but surprisingly none has been able to prove this as yet.

(ii) Fermat’s Conjecture or Fermat’s Last Theorem: For positive integers x, y and z, the equation xn + yn = zn has a solution only in the case n = 2, but it has no solution for n > 2 (n ºN). Ever since Pierre de Fermat posed this problem in 1637, no one has been able to prove this.

Only in 1996, Andrew Wiles gave a proof of this almost after 358 years. Interestingly the greatest mathematicians of this period tried this problem but could not crack.

(c) In Graph theory too, there are some problems which baffled the efforts of many. One such problem is:

(i) Hamilton’s Problem: Is it possible to find a path through the edges of a dodecahedron, which starting from a vertex passes through every other vertex exactly once returns to the starting vertex?

The problem has not yet been solved though many outstanding mathematicians tried this ever since Hamilton proposed this problem in 1859.

Fermat

(ii) The Four Colour Problem: Are four colours sufficient to colour any map so that no two adjoining countries get the same colour? 

This is an over-simplified version of the actual problem. The actual problem could not be presented here as that requires some precise notions of graph theory, yet it is given here to give an idea about the problem and its apparent simplicity.

The problem was raised in a class room in October, 1852 by a student when his teacher was discussing colouring of a map. His question was – What is the minimum number of colours one requires to colour any map so that no two adjoining countries get the same colour? The teacher after thinking for a while said – Five, but I don’t know whether four will suffice. It took more than 300 years to get the correct answer. In the year 1976 an American Mathematician Hawken and a German Mathematician Apple worked together and using the latest generation computer came out with the solution that four colours suffice.

They divided all maps into several categories and tested each category by a computer programme as to the requirement of colours complying with the condition. Almost all great mathematicians of the 19th and 20th centuries tried this problem sometime in their life but to no success until computers came forward to help us for a solution. Incidentally it may be mentioned that there is no proof as yet which does not use computer. In Mechanics, a problem that has sizzled the Mathematical world for several centuries is the three body problem posed by d’Alembert in 1747 and modified by Clairaut in in 1749. Henry Poincare gave a partial solution to the problem in 1987, but the complete solution of the problem is yet unknown.

The Three Body Problem:

Given an initial set of data specifying the position, mass and velocities of three bodies at a point of time, is it possible to determine the motion of the three bodies in accordance with the laws of classical mechanics ?

The above are not the only problems that played an everlasting influence on the advancement of mathematics. There are indeed plenty of unsolved, i.e., open problems many of which can be found in the Scottish Book, The Problem Book of Ulam. Apart from these, the great mathematicians like Paul Erdos, Nagy, Riesz, Schwarz, Tate, Noether and Hilbert are great sources of beautiful problems.

References:

1. Boyer, C.B. (1989) - A History of Mathematics, Wiley

2. Chatterjee, D. (2002) - The Study of Mathematics, Mathematics Today, June, 2002 

3. Chatterjee, D. (2015) - Introduction to Teaching, Atlantic. 

4. Chatterjee, D. (2004) - Mathematical Modelling, Mathematics Today
5. Kline, M.; Mathematical Thoughts from Ancient to Modern Times
6. Plotker, K.; Mathematics in India, Princeton University, N.J.
7. Struik, D. J.; A Concise History of Mathematics, Dover, N.Y.
8. Ulam, S.; The Scottish Book
9. Ulam, S. (1996) - Adventures of a Mathematician, UCP
10. Ulam, S. (2004) - Problems in Modern Mathematics.