A table of numbers, a matrix like a spreadsheet, has rows and columns. Swap the rows with the columns by turning the matrix on its side (or mirror-imaging it about the diagonal) and you've transposed it. Take a complex number and flip the sign of the imaginary part and you've conjugated it. Take a logical statement "If A then B", turn it around in various patterns, and you get the inverse, the converse, and the contrapositive. Exchange the numerator and denominator of a fraction and you have the reciprocal, the inverse of the original number.
We give different names to these transformations, but in some sense they're all forms of inversion. The mathematician Jacobi said "One must always invert!" Turning things around is an extraordinarily powerful tool for creative discovery. Looking from a new angle reveals unexpected truths. Swapping effect with cause, destination with source, or future with past sometimes shows new links among entities. Systems have subtle symmetries which we often overlook. Mirror-imaging, reflecting, trading places, can make those patterns visible.
Thursday, September 02, 1999 at 20:24:38 (EDT) = Datetag19990902