The substance of exchange is or whether the exchange is fair or unfair. Whatever the matter and the context in which exchange takes place, the touch of exchange is not a relation of equality. Finally, if there is nothing mathematically profound about Aristotle’s notion of the touch of exchange, then it is hard to imagine that his more complicated, and even less clear, the geometry of exchange represents any profound mathematical concept. One can also say that Aristotle’s principle of the radix of money, similar to his concept of exchange, has nothing to do with reality. Even though his primordial theory of how coins came into being sounds simple and rational, it is nevertheless fictitious. The first community or family, its the need of exchange, it’s split into parts, the subsequent barter of such things as wine for corn, the development of more complex forms of exchange, the difficulties associated with carrying the necessities of life, the agreement among men to employ useful objects, the stamping of these objects, etc., are all figments of Aristotle’s imagination and have no historical support. Even his reference to the exchange among “barbarians” is imaginary, since we have no evidence of any systematic exchange among these “barbarians” without the use of money. Setting aside the fictitious nature of Aristotle’s theory of the origin of money, one can also say that the story does not even square with his concept of exchange as equality. If systematic exchange is possible without the use of coins, as in the case of the original exchange or the exchange among “barbarians,” then it is not necessary for money to act as the dispose substance of exchange; nor is it necessary, as mentioned earlier, for price to act as the equalizing force.
That is, not only was swap not difficult in the age of commercial revolution, but it was also carried out side by side with monetary exchanges and, indeed, in some cases was preferred to monetary transactions. Furthermore, the middle ages exchange was fraught with fraud and deceit and this made the Aristotelian concept of equality in exchange even more meaningless. To be more specific, in the first cross I review the concept of equality and justice in the writings of the scholastics, particularly that of Thomas Aquinas. Various versions of “just price” are looked at briefly and it is concluded that whatever the interpretation, the just price theory is incompatible with the real market practices as conveyed in the medieval manuscripts. The second section is devoted to explaining the nature and structure of these medieval or “abacus manuscripts.” The third section deals with the “rules of barter” that one finds in these writings. Here, various barter mechanisms are discussed, and it is shown that the lack of arithmetical knowledge can result in one being cheated in barter. Indeed, the more complex the barter scenario, the greater the chance of losing in the trade. The fourth section presents formal models of different barter scenarios. This helps to achieve a clearer understanding of the middle ages exchange and sets the stage for further analysis in future chapters.