Brief history on mathematics

The homework about origins of mathematics has accompanied the evolution of social systems. Many, many social needs require calculation and numbers.  Conversely, the calculation of numbers enables some complex relations and interactions between some people. Numbers and calculations with them require a well-organized operational system. Such systems were perhaps the earliest models of complex rigorous systems. As we will see, not just one but several number systems come to us from antiquity.  However, as interesting the basic notions of counting may be, the origins of mathematics include more than just enumeration, counting, and arithmetic. Thus, also considered are other issues of mathematics to be considered.  Number provides a common link between societies and a basis for communication and trade. This illustrates various entries into our ancient knowledge of how our number system began. Examples range from prehistory to contemporary.

The ancient evidence is easy to accept, but the modern examples yield more conclusions. It is remarkable that even though mathematics achieved its first zenith twenty three hundred years ago, more than a hundred generations, there are peoples today that still count with their fingers or with stones that have no real language for numbers, and moreover have not the general concept of number beyond specific examples. There is every reason to believe that in the future, near or distant, there will exist a far more credible theory for the origin of counting than there may ever for the development of geometric proof, the Pythagorean Theorem in particular.  Another view is that mathematics may have an even earlier basis on ordinals used perhaps for rituals in religious practices or simply the pecking order for eating the fresh game. It is a compelling argument that long before early man had need to count his sheep or cattle, his primitively organized social had simple rituals based on rank and priority. And not just who was first but also second, third, fourth, and so on, likely to most members of every group. Such basic questions are thus:

• Who eats first, second, etc?
• What comes first, etc?

There is ample of observational evidence that socialized animals of diverse groups, say the baboon and red deer, have some rudimentary sense of order of precedence, called dominance hierarchy. The objects of the order or hierarchy are wide ranging also, for example, for grooming, for breeding, for eating, and the like. Later on, the needs of counting and assigning numbers to sets of objects required more sophisticated calculation techniques, i.e. the beginnings of mathematics. There seems to be little direct path from ordinals to actual arithmetic. So, while ordinals may offer a plausible case for first origins, the evolution to cardinals was necessary before calculations became possible. Therefore, we will pursue the cardinal numbers viewpoint.