Beyond Calculus: Motivating the study of Analysis

Embark on a captivating journey into the realm of mathematical analysis with the inaugural video of our enlightening series. Join us as we unravel the fundamental concepts and dive deep into the intriguing world of real analysis. In this thought-provoking lecture, we not only explore the core principles but also provide compelling insights into the significance of studying real analysis. Discover the answers to pressing questions such as "What does real analysis study?" and unravel the mysteries behind the inherent fascination of this field. Don't miss out on this engaging exploration that promises to ignite your curiosity and lay the foundation for an enriching analytical adventure. Subscribe now and join us on this intellectual voyage!
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What are your thoughts on the philosophy of mathematics? Share your insights and perspectives in the comments section below or in our discord channel
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Thank you for joining us on this intellectual adventure into the world of mathematics and philosophy. We look forward to your comments and insights!

[@robertstuckey6407]

I like analysis because its the first uni course that really takes a microscope to mathematics. I love learning how the skyscaper of mathematica is possible because of the microscopic world in the soil at its base.

[@academyofuselessideas]

Thanks for sharing! your perspective reminds me of an analogy that Bertrand Russel used for mathematics... he said that seeing math is similar to seeing the real world. You can see certain range of things but if you want to see things farther apart, you need a telescope similarly of how you can build more and more complex theories to understand involved things... but that if you want to go down to the fundamental levels you have to build microscopes... In this analogy, if calculus is what you can see with your bear eyes, then ODEs, PDEs, probability, stochastic processes, etc are akin to telescopes... on the other direction you would go to Analysis, topology, logic in order to understand the atoms that form all those mathematical structures... I might not be explaining it too well though, but i thought I'd share that analogy!

[@robertstuckey6407]

​@@academyofuselessideas Exactly. Also it didnt hurt that my course immediately gave payoffs like "this is why we can take irrational powers"

[@academyofuselessideas]

@@robertstuckey6407 Great example... amusingly, many people don't think too much about what taking powers of irrational numbers mean!

[@nafisfaisal5817]

This is a great video, all your videos are pretty good really. Can you suggest some more books that go into why we need to study real analysis before delving deep into the topics?

[@academyofuselessideas]

Definitely! as a matter a fact, someone asked about books for analysis a few weeks ago, and I wrote a post in the community section of the channel... So, feel free to check out that post for a more detailed answer. That being said, a good book to start analysis is Jay Cumming's Real Analysis. In the first chapter he motivates the study of analysis... As for motivation, I am currently reading A Radical Approach to Real Analysis by David Bressoud. This book explains analysis from a historical perspective, and poses the problems as they appeared... The book assumes that you have a strong background on calculus and applications, and it goes to some heavy calculations and from those calculations, it motivates the needs for a stronger foundation... Also, if you want to talk more about it, feel free to join our discord and ask!

[@nafisfaisal5817]

@@academyofuselessideas Thanks a lot, I have both radical approach books by Bressoud but unfortunately haven't gone through them. I'll have a look then

[@academyofuselessideas]

@@nafisfaisal5817 Awesome... let me know if you have any troubles reading them! and what you find interesting!

[@stefanalecu9532]

This has been a great video, I enjoyed it. Are you Finnish by any chance? Since your accent is peculiar and I like it

[@academyofuselessideas]

I am glad you enjoyed it... I am working on the next video fo this series in which we will go deeper into what a number is, and what we need for calculus.... I am also glad that you liked my accent but I am not Finnish... Come to think about it, I have only met a couple of Finnish people, and both of them had flawless English!

[@jarupbg]

Trying to make sense of the absurdities..

[@academyofuselessideas]

Thanks for your comment! I like the emoji thingy!