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Yesterday, we talked about one of the most remarkable

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features of mathematics. We have the freedom

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to choose our starting assumptions. But once

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those assumptions are in place, we no longer

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get to choose the conclusions. Logic leads the

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way. Well, today, I want to show you just how

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surprising that journey can be. For many years,

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mathematicians believed that proving the Pythagorean

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theorem, required a particular trig identity,

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specifically sine squared plus cosine squared

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equals one. The identity seems so fundamental

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that it almost feels eternally connected to the

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theorem itself. In fact, many believed there

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simply was not another way. In fact, I have a

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book that is on the authority of the Pythagorean

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theorem. that specifically states that this trig

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identity must be assumed. Then, in 2022, mathematicians

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were humbled because two high school students

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from Ohio found a proof. Check this. While working

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on a school project, they developed a proof of

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the Pythagorean theorem. that avoided using this

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identity altogether. And the proof was carefully

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examined by professional mathematicians, and

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ultimately it was even published in a peer -reviewed

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journal. Think about how remarkable that is.

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The Pythagorean theorem has been studied for

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over 2 ,000 years. There's hundreds of proofs

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that have been discovered. Yet two students,

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Simply by asking, what if we try a different

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approach, found something that many experts did

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not expect. And to me, that is one of the beautiful

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things that math teaches us. Even when we are

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convinced we understand something completely,

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there may still be another perspective waiting

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to be discovered. Now, this does not mean experts

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are unreliable. In fact, it's quite the opposite.

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Mathematics progresses because mathematicians

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are willing to let logic correct even their strongest

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expectations. Humility does not mean doubting

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everything. It means holding our conclusions

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with enough openness that if logic points somewhere

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unexpected, we are willing to follow. Let me

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end with one of my most favorite quotes from

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Paul Lockhart in his book, The Mending of Broken

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Bones, which is a modern guide to classical algebra.

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So here's what Paul says. We thought it was obvious

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that any two lengths must be commensurable. Not.

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We assumed that every number was algebraically

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or geometrically describable. Not. We thought

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we were the lords and masters of mathematical

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reality. Not even close. We get to play there,

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and we certainly get to invent and discover and

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explore. But we most definitely do not own the

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place. In fact, my favorite feature of mathematics

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is that it continually blows my mind and teaches

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me humility. To be utterly and absolutely wrong

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is one of the greatest gifts one can receive.

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And here's how he ends it. Luckily, mathematics

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is extremely generous in this regard.