Sitting in a math class, and the professor announces that the next topic will be a brief study of matrices (matrix is the singular form). Then is asked a show of hands of those who have NOT had some previous experience in the topic. Up goes my hand, relieved to see that mine is not the only uncluttered mind, but saddened that there are so few of us. Those emotions are replaced when the professor announces that he will ‘go slow’ so that we midgets can keep up with the crowd. Thanks.
As he takes us through the steps of ever increasing arithmetic manipulation, the point is made that some properties of matrices are commutative while others are not. It is the non-commutative properties that are of interest, he observes. For those of you who have my level of understanding, note that an arithmetic operation is commutative if the order of the process returns the same result; 3 * 2 = 6 and 2 * 3 = 6.
As the link above reports:
Records of the implicit use of the commutative property go back to ancient times. The Egyptians used the commutative property of multiplication to simplify computing products. Euclid is known to have assumed the commutative property of multiplication in his book Elements. Formal uses of the commutative property arose in the late 18th and early 19th century when mathematicians began to work on a theory of functions. Today the commutative property is a well known and basic property used in most branches of mathematics. Simple versions of the commutative property are usually taught in beginning mathematics courses.
But, predictably, there is a large portion of mathematics that is not commutative. I knew it was just too good to be true. As the professor observed, there are many, many examples in life where the order of a process is very important. As examples, he pointed out that opening the window and sticking your head out of the car window are operations where the order of things is critical.
Wikipedia expands on the idea:
Noncommutative operations in everyday life
- Washing and drying your clothes resembles a noncommutative operation, if you dry first and then wash, you get a significantly different result than if you wash first and then dry.
- The Rubik’s Cube is noncommutative. For example, twisting the front face clockwise, the top face clockwise and the front face counterclockwise (FUF’) does not yield the same result as twisting the front face clockwise, then counterclockwise and finally twisting the top clockwise (FF’U). The twists do not commute. This is studied in group theory.
I’m confused but more impressed than ever with the nature of our existence. How can an idea as powerful as mathematics embrace contradictory behavior? Why do we think that mathematics can explain the physical world when it is riddled with inconsistency? Could it be that the nature of our existence transcends the universe of mathematics?
Am I having a metaphysical moment?