Equations Speak Louder Than Words

We have thus far created a strong link between familiar intuition and formal mathematics, with the intent of constructing a framework with which to better analyse and understand the complexity of computations. We continue on this trajectory by classifying decision problems according to the resources they consume on deterministic (for the same input, will always produce the same output on different runs) and non-deterministic (for the same input, can produce different outputs on different runs) Turing machines. Our resource of consideration will be time, T(n), called time complexity, where n is input length. A similar analysis can be done for space or memory.

Definition 4.1 Let T(n): N → N (T(n)‘s domain and codomain is the set of Naturals) be a proper time function. Then DTIME(T(n)) is the time complexity class containing languages that can be recognized by deterministic Turing machines (DTM’s) that halt on all inputs in time T(n), where n is the length of an input.…

Rise of the Machines II

In my last entry, we introduced a formal definition of a Turing machine (TM). In this article we will look closer at this mental device and see how it works. To begin with, we can examine what a physical TM might look like. I have included a picture from Wikipedia.

Turing Machine Illustration, Wikipedia, (Drawing after Minsky, 1967, p.121)

A TM is made of tape of infinite length, consisting of chain of cells. In each cell there is a symbol which the machine can read or write over, one cell at a time, using the machine’s head. At any given time, the machine is in one of a finite number of states stored on it. It can do basic operations; move right one cell, move left one cell, read, print, erase and change states. By changing from one state to another, the machine can remember previously attained states.…

Beauty is in the Mind of the Beholder
Often mathematicians speak of finding beauty in their subject, it is reasonable to ask what they mean. Of course, one will get as many answers as there are maths practitioners, but I shall hazard a generalized answer here. Mathematicians deal with abstract objects that exist in the mind, and share a suspicion that these abstractions are in fact real. So like real things they have properties, and enter into relationships. The feeling is that if abstractions behave as real things, then they must themselves, at some level, be real. If they are real, then they must also have relations with other concrete things, and indeed, the ultimate aspiration is to develop an abstraction that describes the world. This is the fundamental quality that separates the practice of maths from, say, intellectual games. Our minds evolved to find order in the universe and mathematics gives us as systematic and dependable (to some extent) way of doing so.…

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.…

Tikaverenga nezve matangiro nemakuriro akaita ruzivo resvomhu pasi rose, tino katyamadzwa kuti hakuna zvakawanda zvakanyorwa pamusoro wezvakaitwa nevanhu Africa yevatema. Kushaikwa kwezvinyorwa uku kunopa kuti tifunge kuti hapana ruzivo rwesvomhu rwakakosha runobva munzvimbo iyi. Asi ichi ichokwadi here? Tichitarisa matangiro akaita svomhu tinoona kuti yaive mbesa yekuti vanhu vakwanise kubudirira mutsinhana nemukurima. Makore churu apfuura Africa yevatema yekawona kukwira kwemarudzi avanhu kwakawanda; kukwirira uku kuchibva mutsinhanha nemukurima; Kwakaita vanonzi vaNok vekuma dokero kweAfrica, kwoitawo vaBuganda kumabvazuva, koitawo vainzi vanhu veGreat Kongo vachiri kuwanikwa pakati peAfrica; tisinga kanganwi rudzi rwekwaMutapa vaiva vakavaka Dzimba Dzemabwe; pamarudzi ose awa nemamwewo akakwira nekudonha hapana here rudzi rwakaumba ruzivo rwesvomu, sokuti mamwe acho akanga akabudikira zvakanyanya tichitarisa pasi rose nguva iyoyo. Mubvunzo uyu wakandipa kuti ndiite tsvagiridzo munyaya iyi, izvi ndizvo zvimwe zvakakosha zvandakawana.

Pano ganhurana nyika dzeD.R. Congo neUganda ndipo pane muromo werwizi rweNaire (Nile) unonzi Lake Edward. Makore makumi maviri ezviuru apfuura (25,000 years ago), pane musha wevanhu wakadzika midzi pamuganhu uyu.…

On Friday the 21st of August, I will be giving a presentation on the history of mathematics in Africa. The event will be hosted by the UCT Mathematics Society, in room 212 Mathematics Building, starting at 13hrs00. The talk will briefly trace the development of mathematical practice and thought through out the continent. However, an emphasis will be placed on sub Saharan Africa, as there is little recognition for this region’s mathematical achievements. That not withstanding,  I will include some of the more recognised, though not well promulgated, contributions from other parts of the continent. As an African, I believe it is incumbent upon us to recognise our own achievements before expecting anyone else to. Lets meet and share information. Africa counts!

A quick perusal of the history of mathematics and one quickly gets a sense that sub-Saharan Africa has not made any meaningful contributions to modern mathematics. This ‘absence’ gives a distinct impression that there was very little, if any, mathematics practised in the region during pre-colonial times. But this stands against reason; mathematics in the main was developed by stable societies in order to facilitate trade and agriculture. Over the last millennia, sub-Saharan Africa has been home to numerous enduring civilizations, occupying different regions at different times, and engaged in agriculture and trade at varying extents; from the Nok civilizations of West Africa to the Buganda in the East, from the Great Kongo people of central Africa, to the Mutapa Empire in the South; of all these civilizations, could none have practised mathematics in a form that can be understood today? With these and many other considerations, I set about investigating the history of mathematics in Africa.…