Euclid's Elements has proven itself to be one of the greatest mathematical textbooks to have ever exist. I believe that one of the the reasons why Euclid and his book have been so important and popular for centuries is that it is able to uncover ground without circular reasoning. I speculate that there must have been frustration amongst mathematicians before Euclid when it came to logically proving ideas, despite how blatantly and intuitive it seems. For example, if one were to look at a diagram of a square, how do you know that it is indeed a square? I think that it's the general concept of not taking things for granted, and making sure we always pay attention to our surroundings is what we have valued throughout these years. Also, if I remember correctly, Euclid's fifth postulate raised many eyebrows until hyperbolic geometry was studied. This is a great example to always question the obvious!
There is elegance in the way we prove that things that occur in the world. Is there beauty in defining the given truth? I believe so. It's quite fascinating how we can fix a couple axioms, and then construct complex theorems out of them. Euclid's work was mainly based on the 2-D plane yet it served as a basis for relatively and related ideas like determining the shape and dimension of the universe. The beauty comes from how we can set some simple, intuitive facts (axioms) and develop them into sophisticated and crazy ideas through logical reasoning and proofs.
Fantastic reflection, JJ! You highlighted an insightful point about how Euclid’s work, though primarily based on the 2-D plane, laid the foundation for monumental mathematical ideas. It’s indeed amazing how starting with simple, intuitive axioms can lead to complex and even mind-bending discoveries through logical reasoning and proofs. Excellent observation about Euclid's fifth postulate— a great reminder that even the most obvious ideas are worth questioning. I did an activity that touched on hyperbolic geometry - I will share it if I can find it!
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