A few weeks ago, when I was studying for my Modern Algebra final with Greyson, we read this bit from Judson's Galois Theory, in the chapter where he discusses the irreducibility of polynomials.
Greyson pointed out the wording while we were studying, and we thought it was quite funny- "surrender easily" evoked this mental image of some old white haired British mathematician brandishing a theorem at a Big Bad Polynomial in a valiant mathematical duel. I liked the idea of fighting every math problem, of seeing solving a problem as finding a way to make them yield and surrender.
I am reminded of this excellent comic from Abstruse Goose:
I love that quote. I still remember the first semester of freshman year when I went to my math professor Mu Tao Wang's office hours in a desperate but unsuccessful attempt to make some sense of the mess that was proof based linear algebra in my head. I wanted some advice as to how to study, and how to begin properly learning the material and really understanding it. He gave me some advice that has slowly made more sense as I've studied and learned more math, and he told me that the way to properly read the material was to engage it. The way to learn a theorem is to test it and try it and challenge it, to see under what general cases it holds true, under what special cases it might not, what the edges cases are, what part of the theorem uses which of the given, which conditions are necessary, to grapple with it until you've proven to yourself without a doubt that it is true. In my experience, that's a very important part of learning math. It's easy to accept certain things as true by handwaving and accepting just the right amount of fuzziness to avoid hurting your brain, but I think to properly learn a subject you must fight it. Why is this the case? Can you prove that it is not? What counter examples come to mind, and why are they not counter examples? In my opinion, proper math learning is a very active endeavor- it is all too easy to let things go and accept a theorem to be true, but then you cheat yourself out of real understanding.
I think this also applies to everything we study and learn academically. It is easy to accept certain things as magic, and it sure is easier to prove a theorem if you're willing to accept things through proof by professor or proof by vague recollection of proof in class, but to grok it- you gotta fight it. Take CS as an example. Why does a particular piece of code compile this way? How does the JVM handle garbage collection? How is source code converted to machine code, and what optimizations were used along the way? Why is the runtime of this algorithm O(m log n)? Can it be faster? What type of syntactic sugar is available in a particular language, and in what forms? The new cs student will happily accept that String str = "hello" will initialize a new string in Java, but most don't think about what's really happening under the hood to create the string. (Hint: answer is not black magic.) Similarly, for CC, I've found that the best way to learn is to challenge things. Why does this philosopher think this? What are their assumptions, how did they use the assumptions, and are they wrong about what they assume? In almost anything we can study, we ought not to just passively "learn"- instead, we should fight it.
Actually, to apply this more broadly, I think this holds true for all knowledge and everything we learn and accept.
In IBSL Psych a few years ago, we learned about confirmation bias- a cognitive bias towards confirming our existing beliefs. Apparently, we are geared towards things that support our thinking, and naturally dig out evidence that "proves" our existing biases and opinions. For example, Democrats tend to read news with a liberal tint more often, and similarly for Republicans. Obviously, this is not great- we just end up reading, learning, and remembering stuff that confirms what we already know, and a lot of the times we fail to consider opposing views. In the case of religion as well, I think people would benefit from learning different perspectives, and understanding what those who are different believe in or how they think. Even if you don't agree or see eye to eye on everything, you will refine and strengthen your own beliefs by continually challenging them, by seeing why another might disagree. Barring the sad reality that we are often more wrong than we know, even if we consider an opposing view that ends up to be false, we will have benefited from understanding why it is wrong. I've also found that this is relevant on a personal level to me. I am a chronic over thinker, and I often abuse myself over silly conclusions and concocted scenarios from unrealistic assumptions. I worry way too much a lot of the times about things that are far too little, when often all I need is for someone to prod me and remember me to consider other possibilities, to fight what I naturally assume.
Fight what you know! Nothing deserves to be a part of your foundation of truths until they've been tried and tested. Ultimately, I think the bottom line is that there is a whole lot of stuff out there that is not true or just plain wrong. After all, even if there are infinite truths, there is an infinity of infinite not truths, and it is a rare and unique quality for something to be true, so I think it prudent to not accept things so flippantly. Even if you are right, you will always benefit from fighting what you know and questioning what you think. Perhaps you end up changing your views- that's good! Or perhaps you understand why you disagree with another, and it strengthens your views- that's also good! In learning, beliefs, opinions, views, and thinking alike, whether you end up right or wrong, it is always good to challenge. In my opinion, it is the burden of the person who holds the belief to be able to defend it, and if you can't, then you damn sure better figure out how or at least be open to change. I think if we are a little more critical of what we believe and challenging of what we let in, if we become intellectual warriors, it'll lead to better discourse, fewer but stronger beliefs, and better understanding of things but in particular why we believe what we believe. And those are good weapons for any of us- not just mathematicians.