This may possibly be the best summer session I've ever had since entering college. I've enrolled myself in Precalculus and General Physics, and never before have I felt such anticipation to wake up at 6:45 a.m to drive across the Bay Bridge and back just to attend classes. I feel that the study of physics opens my mind to see so many invisible worlds all around us, and how they interact with each other to manifest the physical phenomena we observe. And math is the language of all that occurs: a strange, mind-bending game which can in fact be conquered and understood. Yet the more I learn the more questions come to surface, and this makes me appreciate science all the more. These 2 concentrations constantly feed off each other to create a beauty so complete, it just makes me want to smile.
So, how do I know my math instructor is an amazing teacher?
The first day of class, while reviewing the standard definitions of natural numbers, integers, rationals, real numbers, etc. I asked him what another example of a non-real number, besides i could possibly be. Instead of answering with just a dry, 2 second reply, he launches into the history of mathematics:
Math was created to model the world around us. Quite obviously, whole numbers exist since the only way we can quantify more than one object is to count: 1, 2, 3, 4... Negative numbers must exist in order to describe being in debt and such. Fractions are highly useful, and irrational numbers occur naturally e.g. the the hypotenuse of a triangle with both sides of length 1 is the square root of 2. But are non-real numbers, well, real? Can we find them out in the world? A debate was held on this, and the conclusion was reached: no number actually exists. There are no numbers that can tangibly be found in the world, there are only well-defined concepts.
This intrigues me, as I have had this quandary inside myself. As many other scientists have also pondered: to what extent do the mathematical workings actually describe reality?
The other day, confused on his requirements, I asked him if he wanted English sentences as proof or mathematical manipulations. His response to me was, "They are one and the same."
Well, now, isn't that something?