Saturday, February 25, 2012

About Number Definition, Pure, and Applied Mathematics

If we agree that math is about counts, and counts only (as it is, since numbers, counts come from the cardinality of sets and set theory) then geometry doesn’t belong to mathematics. It is, by some authors (mentioned in "What is Mathematics Really", R. Hersh), considered impolite to have any geometric drawing in a mathematical text. Geometry has link to mathematics as the morning purchase of vegetables on the local market has. The geometry is only more convenient (perhaps!) in representing numbers and their relationships. Geometrical figures do only one thing to mathematics – by measuring the distances, angles, etc. we generate numbers, and sets of numbers. None of ZFC axioms refer to anything geometrical in the same way that ZFC axioms do not refer to the bunch of carrots at the local produce markets.

Geometry can help to visualize certain mathematical relationships and results. But, the link between pure numbers and sets to the geometry is in essence arbitrary. Geometric interpretation of mathematical results are neither mandatory nor necessary.

For mathematics, it is completely arbitrary what or who generates numbers. The process of numbers selection, generation, numerical operations can be scientific, guessing, or a product of any dogmatic philosophy. Math couldn’t care less. As for geometry, the reason it has a strong presence in mathematics is just because of some of its practical applications. The reason why we can abstract real world into points, lines, planes, spheres is extraneous to mathematics. For whatever reason a line is drawn, and for that matter, what that line represents abstraction of, is not a part of mathematics. From math point of view we draw lines to generate numbers by measuring the lines’ lengths. Measuring process (with instruments, visually, or in any other way) again, is not part of mathematics. Math will see only the number you obtained.

For example, when we write 2 x 3 = 6 (without any explanation) will the reader know where 2 and 3 came from? Of course not. It can be from 2 baskets, each one having 3 apples. Or, it can be from 2 cars, where each car has 3 passengers. Why do you need a rectangle with sides 2 and 3 to explain you this mathematical result? You don’t need it.

As much as apples, cars, are not part of mathematics, in the same way is not rectangle or any other geometrical figure. Geometry is perhaps interesting because it selects, generates certain sets of numbers that are of interest in everyday applications, like lines, squares, rectangles, circles. It is quantification of these figures and their measures that matter to mathematics, and not figures themselves. The thought  process that takes place in defining a circle as an ideal abstraction of all real world attempts to make a circle (as well as a straight line abstraction of all straight directions) is a nice thing to think about, but that’s not part of mathematics. Once you “idealize” circle, math cares only about the numbers you provide by measuring them.

Simply put, no geometric figure should be considered an element or part of pure mathematics because none of the theorems in math are proven using them. If seemingly geometry terms are used in proofs or appear to be a focus of study, like trigonometry or differential geometry, it is because the axioms of geometry are part of it, but, they are not part of mathematics. Mixing ZFC axioms and geometry axioms is like mixing ZFC axioms and axioms of any other system, including "marbles used in counting", carrots methods of purchase, quantitative finance rules, etc..
"The formalist makes a distinction between geometry as a deductive structure
and geometry as a descriptive science. Only the first is mathematical. The use of
pictures or diagrams or mental imagery is nonmathematical. In principle, they
are unnecessary. He may even regard them as inappropriate in a mathematics
text or a mathematics class."  ("What is Mathematics Really" Rueben Hersh)

Of course, it doesn't mean you should not use them to better communicate your ideas, investigate new directions in math or other sciences, or visualize a bit more difficult concepts in mathematics.But, you have to clearly differentiate between mathematics and these non mathematical objects and concepts.

Mathematics is defined not by objects it counts, nor by reasons or logic why those objects are counted, but with concepts used to define mathematics axioms and to define proofs of mathematical theorems.
[ to be continued...]

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