Motivation: Mathematics for the Masses

It is my firm conviction, and I preach it when ever I can, that one day in
the near future, mathematics shall save us all. A ”grand claim,” I hear you
say; but not at all, mathematics is believed by many to be the language of
Mother Universe, and indeed, those who have adopted it as a native tongue
have been granted glimpses into her secrets. Intuitively, my claim is not hard
to defend, given the pervasive influence that technology has over our lives; from
health to communication, entertainment, art and culture; science has become an
indispensable companion. Amongst the sciences, mathematics is the common
denominator that binds them all. It is the life blood of all other scientific inquiry.
As the world faces seemingly intractable challenges, be it global health, world
peace or universal prosperity, it has become imperative that more of us engage
in scientific exploration and innovation, for, if history is anything to go by, this
is where we shall find the answers we seek. I propose that mathematics be
declared an international official language! The idea is that the more people
understand and are comfortable with mathematics, the more scientists we will
have, which means more minds pondering the mysteries of the universe. That
can only be a good thing for us all.

In many ways mathematics is its own worst enemy, for the thing that has
made it powerful has also served to weaken it. I say this because one reason why
mathematics continues to grow in influence is because those who have grasped it
continue to grow in their understanding. On the other hand those who, for one
reason or another, missed the boat are left further and further behind. Mathe-
maticians speak mathematics to other mathematicians, and become more fluent
for it. But the more fluent they become the less they are able to communicate
mathematically with society at large, and consequently the less their brilliance
serves where it is needed most, that is, inspiring more people to engage in maths.
In the end, we produce far fewer mathematicians and scientists than we need.
So, what shall we do? Well, we could start by creating spaces where mathe-
maticians and non-mathematicians can come together and interact, specifically
for the purposes of exchanging mathematical ideas. Those of us involved or in-
terested in the subject, at all levels, must make a concerted effort to create such
spaces and reconnect mathematics with the broader society. It is my intention
to use this platform to do just that in my own small way.

In the following months I will be publishing mathematical ideas in computational
complexity. This subject is special because it brings together science and math-
ematics in a way that is more accessible than most others. Though it is formal
and rigorous, it remains palatable and applicable. It is my hope that the reader
will find the material posted here informative and interesting. To start off, I
will delve into the ideas of computability and complexity, to ensure that going
forward we develop a strong sense of what these terms mean in a mathematical
sense.

Introduction to Computational Complexity

What is a computer? On page 173 of a 1927 publication by J.B.S Haldane
entitled P ossible W orlds and Other Essays, is the following entry;

Only a few years ago Mr. Powers, an American computer, disproved a
hypothesis about prime numbers which had held the field for more than 250
years.

How strange, a human being here referred to as a computer. In fact, the term
was originally used as a job title for persons whose occupation involved per-
forming repetitive calculations required to produce such things as tide charts,
navigational tables, planetary positions and other complex phenomena. With
the mechanization of calculating, the term came to be used for any tool used
to carry out mathematical computations. So from this we can deduce that the
word compute can be synonymous with calculate. The subject of computational
complexity, is concerned with computabilty: which refers to problems that, in
principle, can be solved by a computer, which is the same as saying problems
that can be solved by the mechanical application of a finite number of steps, or
an algorithm; complexity is concerned with the amount of resources in time and
storage needed to solve computational problems. As it turns out most problems
we might want to solve are computable, its really a matter of how efficiently we
may be able to carry out the computation. The greater the resources required
to solve a problem, the more difficult that problem is regarded. Putting it all
together computational complexity is a branch of mathematics that focuses
on classifying computational problems according to their inherent difficulty in
solving, and relating those classes to each other.

But is this important? Is this a matter worth pursuing? Is this subject
even really mathematics? The answer to all three in ”yes”. For computer
scientists to be able to answer questions like ”Is this the best algorithm to solve
this problem” or ”Can this problem be solved efficiently” they have had to
rely on mathematics. Mathematics allows for formal definition of the intuitive
ideas presented in computer science, which enables the use of mathematical
tools such as theorems, lemmas and proofs to guide our understanding. As a
consequence, a great deal of its subject matter is theoretical. That said, like
all sciences, mathematical inquiry must be allowed to drift where it may, as
long as there are scientists interested in the question; for as they wonder in
uncharted waters, they stumble onto the very ideas that drive technological
advancements upon which our well being depends. After 1970, computational
complexity made astounding discoveries in a variety of areas which include
public key encryption, NP-completeness, novel types of mathematical proofs,
and the theoretical foundations of machine learning and quantum computing.
Beyond this, the mathematics we will explore, and the skills we will cultivate
are in their own right beautiful and worthwhile.

In my next article, we will look more closely into the notion of computation.