WEBVTT

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Hi, so far this week we've talked about submitting

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ourselves to logic, allowing our intuition to

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be corrected, and admitting that we make mistakes.

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So today I want to talk about another kind of

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humility. The humility to recognize that other

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people may see something that you do not. Now

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one of the beautiful things about math is that

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there is often more than one correct way to solve

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a problem. So two students can arrive at the

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same answer, but using completely different approaches.

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One might use algebra and another might use a

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graphical argument, for example. Someone else

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might recognize a clever substitution that everyone

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else missed. Now, sometimes one method is more

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efficient and sometimes another method might

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give deeper insight. And sometimes a method that

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seems longer. actually teaches you more about

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why the result is true. And that is one reason

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why I really enjoy looking at other people's

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solutions. Not because mine is necessarily wrong,

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but because theirs might reveal an idea that

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I had not considered. So when we're first learning

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math, it's easy to become attached to our way

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of doing things. After all, it is the method

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that makes sense to us. So when we see someone

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solve the problem in a different way, our first

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reaction can be, why would they do it that way?

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So maybe a better question is, what do they see

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that I do not? And that question can change everything

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because it turns another person's solution from

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something to maybe criticize into something to

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learn from. And this is one of the reasons mathematics

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has progressed so rapidly. Every generation builds

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on the insights of those who came before. Different

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mathematicians bring different perspectives.

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They bring different tools and different ways

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of thinking. No single person sees everything.

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And of course that is okay. Humility reminds

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us that our understanding is always incomplete.

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There is almost always another viewpoint that

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can deepen our understanding or simply reveal

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a simpler way to think about a problem. And this

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is a lesson I've had to learn the hard way. And

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of course, not every approach is correct. Mathematics

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still demands logical consistency. Different

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methods must ultimately agree with the underlying

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truth. But when multiple correct approaches exist,

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we do not have to defend our favor as if it's

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the only valid path. Instead, we can appreciate

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the richness that comes from seeing the same

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truth from different angles. So, when someone

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creates a solution that is different from yours,

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here are some questions you can ask yourselves.

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Why do they choose that approach? And what does

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their method make easier to see? And also, what

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can I learn from it? Humility is not just admitting

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when we're wrong. Sometimes it is recognizing

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that even when we are right, we have plenty that

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we can learn.