The Math I Have Learned So Far at UCSB

It might seem to be pretty strange that I was taking 4 math classes last quarter but almost did not do any computation. Contrary to our common belief, math major students, expect applied math major students, hardly do any computation. If you compare my homework to that of a physics major student, you would find that my homework is more like an essay, full of words in the proofs, while the homework of the physics major student is filled with computational equations. One thing that I have learned since I came to America that I would like to share is my experience with math.

No matter what braches of math nowadays, they all share a common trait, called the axiom system. In the axiom system, certainly, there are axioms. In math, axioms are statements that we assume to be true, because they are self-evident to be true. In another word, axioms are so obvious to be true that we just accept them as our starting point of our proofs. For example, in the Euclidean geometry, axioms are statements like “That all right angles are equal to one another,” and “To draw a straight line from any point to any point.” If there were no axioms, then there would be no starting point of reasoning and our proofs would not exist.

So what should we do with the axioms? To deal with the axioms, we need logic. Logic is connection between statements and statements. With logic, we can deduce different theorem and propositions from the axioms we have. Then, we are now able to construct our math system.

It’s pretty interesting to find out the math I learned in different periods of my life different from each other. In primary school, the math I learned is more likely to find patterns, which is totally not rigorous. In middle school, I start to learn geometry and that’s the first time I start to do proofs under axiom systems. However, at that time, the math I learned is from thousands of years ago, and there are many gaps in those math systems. Now in college, I could finally start to learn “real math”.