I am honestly quite impressed that Euclid was able to construct the Elements without any proofs, but after reading a bit more, I was a bit disappointed to see that this book is just a compilation of knowledge, rather than him making discoveries in math like I assumed. So it feels wrong to call things “Euclidean math,” since he didn’t discover many of the axioms and such. However, I think the Elements still serves as an important keystone in math, since without it, we wouldn’t be able to organize the math we know today.
I think it’s important to acknowledge that his work helped us understand the essence of math. Without him, our theorems and axioms would be all over the place, and we wouldn’t even have fields such as number theory and geometry, so I think he deserves a lot of credit for it, even though he is just standing on the shoulders of giants. However, despite all that, Euclid is able to show (rather than proof) his theorems and axioms through geometry and exhaustion, which I think is a beautiful thing. Just like the presentation that Sahl, Brandon, and Caris did on the truncated pyramid volume, there is something beautiful about a visual proof. I think beauty is defined as a subjective pleasure to the eye, so as a mathematician, the simplicity of a visual proof is quite a beautiful sight in the sea of words that are typical of the proofs that we have done.
Thanks for the thoughtful reflection, Leon! While Euclid’s Elements isn’t solely his own work (as you've seen in this course, we sometimes might have given too much credit to Western mathematicians), it’s undeniable that his contributions were foundational. As you said, thanks to his work, we have a clearer structure for the theorems and axioms that shape our understanding of math today. Again, it’s another example of how the development of the field of math is a collaborative effort.
ReplyDelete