AN APOLOGY TO MYSELF

The Wolfram Demonstrations that have come out relatively recently are worth checking out.

I think they could help many of us with mathematizing our understandings or would it be better to say that Wolfram Demonstrations could really help us develop our way of looking at things more mathematically so we can begin to see patterns an relationships that evaded our understandings previously.

What is Mathematization?

What is mathematization? One example may help us work out some sort of answer to that question. It is the example of counting things. We count our children, the number of quarters in our hand and the number of anything we want to think about more precisely, more clearly and of course with counting numbers we get to quantify things, do arithmetic.

We can imagine a pile of nails that looks two or three times larger than another pile of nails.  Assuming that the nails are the same in both piles and that they are packed about the same we can readily conclude that there are more nails in the bigger pile. Of course “bigger” and “lesser” are also mathematical ideas but the counting of the nails in each pile can further mathematize the problem and we can finally conclude exactly how many more nails are in one pile than the other.

So the mathematization of a problem or area of study consists of applying mathematical ideas to that problem or field so as to think more precisely or clearly about things.

Economics or economic activity somehow assigns numbers to things for sale. This makes possible a rather complex economic system, sort of a huge economic game. The rules of the game may not always be clear but numbers are somehow assigned to costs and revenues and all sorts of things and services whatever the exact rules of the game turn out to be.

It seems to me that economics has become something of a science and something of a mathematical architecture or mathematically concocted machine or meta-machine. In other words, mathematics itself shapes economic activity as well as describing it.

One definition of mathematics that is that it is clear thinking. There seems to be much merit in this definition. One is tempted to add, however, that mathematics is clear thinking plus the mental tools used to clarify our thinking. So mental tools or structures are created like number systems which make sorting out things and ideas about things much more precise.

Mathematical thinking can recursively apply this or that package of clear thinking to a particular problem.

Mathematical thinking can recursively apply a succession of mathematical tools, premises, theorems, definitions, to create a huge concoction of precise thought. Whether any human can follow the argument and understand the subject matter is another question.