Here is a sample of algebra problems we do not need.

A and B working together can do a job in 6 days. Working alone, it takes A 5 days longer than B to do the job. How long does it take B to do the job?

A and B are partners in business and their respective share of the profits of the business are in the ratio 5:7. The profits for a certain year were $53,473.75. What was each man's share of the profits?

A man rows 12 miles upstream in 4 hours. He rows 20 miles downstream in 2 hours and 30 minutes. Find the man's rate of rowing in still water and the speed of the current.

Alice is 8 years older than her brother, and 5 years from now the sum of their ages will be 63 years. How old is each now?

Solve

Syncopated Algebra

Rhetorical algebra is verbose and inefficient. So the Egyptians discovered syncopated algebra, a language where the names of numeric variables are abbreviated by mnemonics. Rhetorical algebra stayed, but some components were replaced by the new algebra.

Descartes in the sixteenth century introduced his symbolic algebra, where the numeric constants were designated by the single letters a, b, c, ..., and the unknown numerical variables by x, y, z, .... This represented a major improvement for the physical sciences and erased the use of the old style of math. But as to the world of nonscientists, Descartes lowered an iron curtain between humans and mathematicians.

Euler versus Diderot

Denis Diderot (1713-84), the great French encyclopedist and philosopher, completed his work on his celebrated Encyclopedie in 1772, and left Diderot without a source of income. To relieve him of financial worry, Catherine the Great of Russia first bought his library, then appointed him librarian on an annual salary for the duration of his life. Diderot went to St. Petersburg in 1773 to thank her for her financial support and was received with great honor and warmth. However he caused some problems for Catherine.

Catherine was a great admirer of the French Enlightenment and corresponded extensively with Voltaire, Diderot, and a number of other French writers, but got fed up with Diderot's efforts of trying to convert her courtiers to atheism. She could not send him back to Paris, she had to find some other means to calm him down. Unfortunately, there was nobody in her Court that could stand up to Diderot's genius. But then she had an idea.

Catherine, as a benefactor of science and letters, in 1766, invited Leonard Euler (1707-1783), the most prolific mathematician in history, "Analysis Incarnate," to be her "royal" guest. (Euler's collected works are still under preparation. When completed, they will contain 72 volumes.) She recalled that Diderot, as a man of letters, was not up on science and math, and his coeditor, Jean Le Rond d'Alembert (1717-1783), French mathematician, philosopher, and writer, worked on those subjects. So she asked Euler to cook up some mathematical gobbledygook to prove the existence of God, then asked Diderot to appear and respond to the proof. In a full court session, Euler advanced on Diderot, and solemnly declared:

"Sir, a + (b sup n / n) = x, hence God exists; reply!"

As Catherine suspected, Diderot knew nothing about algebra and was dumbfounded. Great laughter followed the silence and Diderot was so humiliated that he asked for Catherine's permission to return at once to France. She graciously consented.

It is hard to understand how an intellectual giant like Diderot did not ask Euler to define the terms in the formula. Charles Snow must have been right when discussing the chasm between scientists and men of letters. However, things are changing.

The impact of science on everyday life, the emergence of the computer as a tool for millions, has created a new world for the general public. The new users of math are spreading the word, and understanding for math is increasing.

Calculus

Finite Math

Compound statements (Includes tree diagrams)

Sets and subsets

Partitions and counting

Probability theory (Includes stochastic processes, decision making, expected value, Markov chains)

Vectors and matrices

Linear programming and the theory of games (Includes convex sets)

Applications to behavioral science problems (Includes computer simulation)

This was, of course, 1956. Today we need to drop some of the topics and introduce new ones. The most important topic is the math of computers, of finite state machines, automata, and general purpose Turing machines.

The chapter on applications is to be totally revised, spreadsheets providing the basis of applications.

Illusions About Math Required

And I did. Of course, I was not alone. The pioneers all had a solid training in math. This was the heyday of linear programming. Business Week, Harvard Business Review, and others played up the new tool for management. But as the years went by things started to get sticky. For one thing, management could not assimilate the math. Another thing, math took on a life of its own; it became l'art pour l'art. Finally, people started to question the role of math in OR/MS. What math is really required?

Authors of our textbooks have been ambivalent about math. Often it is said that the student does not really need any math. One of our wits in an introductory management science text said that his students will be required to know the following (but in dealing with inventory control the text uses partial derivatives!):

1. How to add and subtract.
2. How to multiply and divide.
3. Know the left from the right.
4. That one number is bigger than another.

Math for Decision Making: Object-Oriented Algebra

In math for decision making we need to deal with a much broader range of objects and their properties. For example, consider FCAST, a mathematical object, the forecast for the XYZ corporation. One component is the sequence of integers representing the five years of 1998, 1999, 2000, 2001, and 2004. There are 15 more integers representing the three forecasts: optimistic, normal, and pessimistic, in thousands of dollars. The object also contains pictures (graphs) of the three forecasts in blue, red, yellow. The single object, FCAST, contains many sub-objects, with a complex set of properties.

The classical math function may deal with many input variables and a single output variable. Decision making involves procedures, a vast generalization of functions. A procedure, for example, a sub-object of FORECAST, may have many matrix inputs and outputs. A procedure maps a set of input objects into another set of output objects.

Millions use computers for financial work, and the software is based on object oriented algebra. While people hate classical algebra, the object oriented algebra of software is acceptable. Spreadsheet users work with procedures, functions, matrices, graphs. Data base users work with alphanumeric variables, matrices, and relational functions.

Computer people have recognized the need for a language to work with many different objects and developed the concept of object oriented programming. I believe that the foundation of these programming languages is mathematical, and for this reason have coined the expression object oriented algebra as the type of math we need for decision making.

Wrap-up

• The objects of math should be vastly extended to include the needs of the sciences of the artificial.
• The style of algebra should be syncopated.
• The opening focus should be on the sciences of the artificial and not on the natural sciences.
• The approach of the infinitesimal should be replaced by Finite Mathematics.
• Stochastic processes and simulations should be stressed.
• Computers should be used as the principal educational tool.
• Applications should be from public service.

We need a specific list of topics. A good way to start is with finite mathematics. We need to expand the coverage of vectors and matrices to include spreadsheets. To expand the discussion of tree diagrams. Cover functions, inverse functions, matrix functions, procedures, that map objects into other objects, spreadsheet functions, algorithms.