This is not a question. It is a demand. I am planning to self study Real Analysis in 2024. Please don't call it pointless; because it has. In fact, it has many.
@@AutoDisheep I am glad that you find real analysis useful! and in the end, isn't that what matters? because useful and useless are fairly subjective. I have little use for a fishing pole but in the hands of an experienced fisher-person it is a fantastic tool! Not only that, but usefulness also depends on your goals. If my goal is to learn russian, i have no use for a french-english dictionary. I hope that the videos help you in your journey (i wanted to make a whole introduction to real analysis series but I got distracted). As you advance, you'll notice the ironic nature of the "uselessness" claims... However, one of the main points is that there are things worth pursuing out of aesthetic value instead out of usefulness (at least as people usually define usefulness)... Anyways, I wish you the best in your real analysis journey... if you have any real analysis questions (or any other question) feel free to join our discord server... one of the main issues with self studying real analysis (or advance math) is that most books don't give much of a motivation for why the topics are relevant, or interesting. It is much easier when you have people to talk about the things you are learning... I can also provide you with a list of recommended books and topics depending on your reasons to study the subject!
@@academyofuselessideas I already have several analysis book that I will dive into. However, if you would be kind enough to recommend me the best analysis book, and tell me why you like it.
@@AutoDisheep It would be easier if you tell me which books you have, so I can tell you about their strengths and weaknesses, and even discuss what problems are good from each one of them. However, I wrote a more detailed answer to your question as a post in the community section of the channel. As I was writing my answer, I realized that it was a bit too long for a youtube comment. Moreover, as a community post, some other people in the channel would be able to expand and give their own perspective on what is the best way to learn analysis. Please check that post out (or let me know if you cannot find it!)
True, some theorems in analysis have found interesting applications (even some theorems that go beyond calculus)... it might be nice to make a video on theorems that turned out to be applied to some real life problems! feel free to share some examples!
Thanks for your encouraging words... Most of my real analysis videos are fairly fast passed since they were originally intended as quick reviews more than for learning the topics for the first time... I might do some more detailed videos in analysis at some point though... in any case, if you have any question about analysis, or any type of math, we would love to talk about it in our discord server... i hope that your previous professors didn't discourage you from learning math or analysis in particular... if they did, forgive them as they didn't know what they were doing (and it is likely that they had the best intentions)... Thanks again for the encouraging words!
i had him as a teacher, and indeed, I learned a lot! Buena, Pedro!! Percibo muchas mejoras con la entonación!! Siga adelante con ewtos buenos contenidos!
@@seebaastian Gracias por el alentador comentario. Es interesante saber que de algo sirvieron mis clases! Tambien me alegra que haya encontrado el canal Mucha suerte con todo y un cordial saludo!
I still mostly care about applied, but a) I'm not comfortable with just accepting the tools as they are without fully understanding them, or b) I'm not sure how I can ever have any confidence in any derived result without having confidence in the starting axioms, and for some domains, something like the results of calculus are the starting axioms... Confidence may be the wrong word, I'm sure other people have checked them and that they work, I just don't know how I can build something that means something to me, out of things I might not be able to unambiguously interpret. That said, I am happy sometimes with a hand-wavy/intuitive notion (at least to start with, though it wouldn't hurt to get better at formalising and checking these). I also like how you describe an almost engineering/computer security approach to reducing dependence on axioms here, as in reducing the amount of moving parts/points of failure, or the attack surface, after all, we want our maths to be secure/robust to logical attack.
yes! some people are comfortable using some results and intuitions, but others prefer to have a very solid foundation... in terms of the turtles analogy, some people are fine standing on the first turtle while others want to go deeper and deeper until they find what is standing in the last turtle... going deep down is quite challenging though, even for something like analysis, it is not that trivial to find the correct axioms... even some people who teach analysis don't go all the way down (and usually in a first analysis class no one goes all the way down because it would be too hard and overwhelming).... But at the very least, we can go down one turtle at a time
My opinion: Well for sure, we have to accept logic, otherwise no statement can be made and it is impossible to anylize the world. We cannot even deny logic without using it. After that, for axioms in other mathematical fields, we create them so that they can be a reliable way to describe some concept found in physical world, then using logic we can derive other scenarios and be 100% sure that they are as much reliable. (Which is where i seek confidence) For more abstract parts of mathematics, we use the same laws that we already have and play with them further, so that they are still logically correct even if they no longer have basis in the real world.
@@MilkywayWarrior1618 pretty solid... we can even question logic, or using different logics to produce some other types of math... for example, you can deny the law of excluded middle (why? there are good reasons for denying that axiom... ask more if interested!) and work in intuitionistic logic... many of the results will agree with those of classical logic but you will need to work harder to find proofs. but in general, i think that math started as an attempt of explaining physical phenomena, and that someway some how, humans realize that those mechanisms of reasoning applied as well to abstract objects. Thanks for sharing!
@@MilkywayWarrior1618 Pretty tough question that gets to a very deep part of the philosophy of mathematics. The example I gave is probably one of the most common challenges to classical logic (classical logic is just the name of the regular logic that you are probably aware of with propositions that you can form from some basic proposition and by using conjunctions (and), disjunctions (or), negation (not). in classic logic, the idea is that you can determine the truth value of each proposition (telling if a proposition is truth or false), and if you know the truth value of every basic proposition, you can find the truth value of using the "rules of logic"... One of those rules of (classical) logic is the rule of excluded middle, which says, that for every proposition p, you have that p or not p must be truth. In intuitionistic logic, that statement is not necessarily truth. The reasons for that are philosophical. A way of thinking about it is that in intuitionistic logic, propositions with truth values are propositions for which you can construct a proof. The fact that you cannot construct a proof for p does not mean that you can construct a proof for p... So intuitionists reject the law of excluded middle which makes their proofs often more complicated than the proofs done in classical logic... I am oversimplifying here a lot for the sake of the explanation, and there are things that i am not being 100% accurate here just to get the general idea across... You can find more on intuitionistic logic on mathoverflow... or if you want to talk about it more with me, feel free to join our discord server... This gets very tricky, and my short explanation here might not be the best one but I hope it shed some light, or at the very least, I hope that it is though provoking!
The very first part of Euclid Book I is a list of 23 definitions. Then 5 axioms and 5 'common notions', which really act like axioms. Book II adds two more definitions and so on. I am not sure why you think the axiomatic method doesn't use definitions - as it truly does. Also - there are several equivalent definitions of the real numbers, for example, as Dedekind Cuts.
Pretty great observation... I should have going deeper into what i meant. If the goal is to provide foundations to calculus (or to any theory) you can take two different approaches. One is to provide consistent axioms that describe the theory (for lack of a better term, let's call this the axiomatic approach). The second approach is to reduce your theory to a simpler theory that is more "solid" (again, for lack of a better term, let's call this the constructive approach)... This approaches are not mutually exclusive but they are more like a rough sketch on how to approach theories. Mathematicians believe that all mathematical theories can be reduced to (or constructed from) set theory. Calculus is not different. You can reduce everything we say here to a conversation about sets. But set theory itself is describe in terms of axioms (usually Zermelo-Fraenkel axiomatic system, but there are other options... and there are indeed other options for a foundation than set theory but such discussion would take us too far). The approach here is axiomatic (as opposite to constructive) in the sense that instead of creating the natural numbers from sets, the integers from the naturals, the rationals from the integers, and finally the reals from the rationals (via Dedekind cuts or equivalence classes of Cauchy sequences (which by the way, we made another short video sketching that construction)), we start directly with axioms for the real numbers (namely, we start with the axioms for a complete ordered field)... Of course, even if we have done all those constructions, one could ask "how do you construct sets?"... Here the analogy of the infinite regression of turtles applies again... The existence of sets is a very deep question in the philosophy of mathematics (the so called question of the ontology of sets). It is pretty interesting but something that most practitioners don't care too much about. Indeed, most practitioners don't even care to reduce math to set theory. They are happy knowing that such reduction is possible in principle. When I argue that we don't "define" the real numbers, I should have been a bit more explicit, and perhaps a better wording is to say that we do not "construct the real numbers" from simpler mathematical objects... When I say that the axiomatic method does not define objects, perhaps a better wording would have been that "the axiomatic method does not tell you what an object is but it tells you how it behaves"... This is a more modern view than the one in Euclid's elements. Euclid probably thought that he was indeed defining points and lines... but eventually mathematicians realized that his definition didn't really say what those objects were, it only said what they do, and gave a possible interpretation for that object. Probably the most influential mathematician to reevaluate the work of Euclid was Hilbert. Indeed, most modern treatments of Euclid's Elements are based on Hilbert's version (this is because in perspective, Euclid's original work was not as mature as how we humans understand math nowadays). All this points are very subtle, and most likely I am not explaining them that well... So, perhaps, an example might help... The first definition on Euclid's elements reads: "A point is that which has no part"... Does that really say what a point is? perhaps, but as you read more and more into Euclid's elements, you kind of realize that such definition does not play much of a role. It definitely serves to help the reader have an interpretation of what a point is, the interpretation helps to guide the other axioms, but if you accept the axioms, and the rules of logic, you will accept the theorems that Euclid proves even if you don't interpret what a point is in that way... indeed, you can find other interpretations of "point" that have nothing to do with the "standard" interpretation... it is kind of crazy, but that was Hilbert's realization (he was one of the first one to give a different interpretation of point and line!)... Anyways, this comment is super long and probably very confusing... But if it piques your interest and you'd like to talk more about it, feel free to join our discord server! I'd love to hear more about your opinion since you raise a very interesting point... Moreover, I am in my own learning path, and the best way to improve one knowledge (in my opinion) is by talking more about this ideas!... Thank you so much for your insightful comment and for kindling the thoughts that I am sharing in this answer!
@IsomerSoma Thanks for sharing that... you might enjoy this series video on topology and continuity... Indeed, topology is the preferred way to look at continuity however coming up with the concept of continuity without any examples in mind is pretty hard. I don't even know if I could come up with the idea of topology on my own... but we have the benefit of our previous generations studying this problem and distilling the idea of a topology for us... Mathematical concepts come from human minds, and I find it fascinating how we figure out some of these abstract ideas!
Definitely! The motivations for linear algebra tend to be "easier" in a way... if you want an answer soon, feel free to join our discord and ask there, I'll give you my answer... or join one of our streams and ask... it might not be as polished as a video but I should be able to give you my perspective on the spot!
@@academyofuselessideas Thanks! I just like the way you present the topic in the video. By the way, I'm self-studying linear algebra, and I'm just curious about any intuition or history or how it evolved that kind of stuff other then the proofs in textbooks.
@@user-ih2ue4bw5q It's nice that you are selfstudying linear algebra. What materials are you using for that? If you are on discord, we can discuss this more... many books don't give enough motivation and you can easily get lost on the abstractions. Linear algebra is a great topic because it is formal but you have tons of examples and applications, so with good materials, you shouldn't have troubles staying in track... I can't think of any textbook based on a historical perspective, but if I come up with one I'll let you know... I'll also ask over discord to see if someone else in the community has a suggestion... I might also ask it in the community tab of the channel!
Enjoyed the vid. Two things: 1) Why dont you start with the ZFC Axioms instead of the Peano Axioms? ZFC are the Axioms in a ton of other math branches. Therefore analysis results can be used in other branches, because they are built on the same axioms. (and vice versa) 2) Imo you missed one important advantage of axioms. (and why a solid foundation is absolutely necessary) Rigour Everyone is agreeing on the axioms and is therefore starting with the same rules. theorems can be checked! Axioms provide as much rigour as possible. Imagine everyone is just starting with different assumptions. Also one should keep in mind why rigour is so important: Logic tells us, that a wrong statement can imply any other statement. (Trump is female => im 3m tall) is a true statement. Thats kinda scary if you consider how long those implications-chains go starting by the axioms. One Mistake and everything can be concluded. Do I really wanna sit in a plane which is based on flawed math in which everything can be concluded? Hell na
thank you so much for sharing! great points. I'll take your suggestions into account if I make a longer version of the video... and I might even go further on a discussion of what's the best foundation for all the things we do in analysis... Originally, I thought of doing something like that, or do a series called "Analysis done wrong" which would start with subsystems of second order arithmetic and use only the axioms that you really need to prove some important analysis results (in the style of the "reverse mathematics" program)... But I realized that such style is mostly of interest of people deeply concerned with foundations (and that's why i wanted to call the series "Analysis done wrong" because it wouldn't be a good series for beginners). There are some issues starting with ZFC... the main problem is if you go that deep, then you probably need to start with first order logic, and that would be a lot of work if the intention is to learn analysis... That might be why building the naturals from sets is often left for a course in Set theory than for a course in Analysis. But it would've worth to mention that all modern mathematics share ZF as it's foundation... as its ultimate turtle... Your second point is pretty great... obviously rigor is a pretty good argument but you also mention something that to me seems as important and worth mentioning... having a common ground and language... that is, mathematicians know what kind of things are acceptable as part of a proof and an argument... Great point. Thanks for taking the time to share your opinion... it is pretty insightful and thought provoking
I never thought about it but that's a little funny. It reminds me of a Bertrand Russel observation. He notices that studying math is similar to seeing things. Humans have a natural range of vision for which they don't need much effort seeing things, but if they want to see very large things far away they need telescopes, and if they want to see very small things they need microscopes. So Russel says that math is alike, there is a level of math that people can naturally address (in this example calculus), but if you want to see far you need to develop more and more tools (like differential equations, functional analysis, etc), and if you want to see smaller things you need also to develop tools (like real analysis). I guess that's why it is called advanced calculus, because it is harder to see than calculus even though it lays its foundations, but that's just my 2 cents. Your observation is very funny and insightful!
I never really understood calculus until I read a book on nonstandard analysis which introduces hyperreal numbers. For an engineer, it is much more intuitive. I really can't understand why schools still start with limits to build up calculus when hyperreal numbers are so much more intuitive. And as a bonus, it has been proven as mathematically rigorous as real analysis. Guess teachers are stuck in their ways just like most of us.
The axiomatic method of proofs is inferior mathematics, just makes math more difficult to understand without adding any practical value to it. You said it yourself in the beginning of this video, real world applications don't require real numbers nor axioms.
Very interesting take... It's pretty hard to think of any modern mathematics that is not based or inspired or influenced by the axiomatic method. Which makes me realize that I don't understand your meaning of the word "mathematics", which in turn makes me realize that I don't understand your meaning of the word "inferior". To make sense of inferior, you need a criteria: A fish is inferior at flying than a bird, but a bird is inferior at swimming than a fish. I do agree that, depending on your goals, the axiomatic method makes mathematics more difficult to understand. And I do think that if you don't see its value, then most likely you are better off not worrying about it. I would say that part of the problem is that the axiomatic method is so widespread in modern mathematics that mathematicians often don't even bother explaining why and under which circumstances the axiomatic method is necessary. Unfortunately, that is a mistake that I have made myself here. This gives me an idea for a video on "why mathematicians bother with the axiomatic method" but a short answer to that question is that mathematicians don't have any other alternative since deep down, whatever idea you construct comes from some beliefs (whether you are conscious of those beliefs or not is a different question). Mathematicians make their beliefs explicit in the form of axioms. Pretty interesting and insightful and thought-provoking comment... I have some other thoughts but I need to learn to be more succinct in my comments!
@@academyofuselessideas 1) Formal math is based on the belief that deductive proofs are infallible. This is why you start by assuming a system of axioms and then deduce theorems from those axioms. But, deductive proofs are not in reality infallible - those axioms could be wrong (which means that your theorems deduced from those axioms are wrong). Formal math fails because humans make errors in deduction (example - the best chess players often make mistakes in deduction). Empirical proofs are not allowed in formal math (which is why it took hundreds of pages for burtrend Russel to prove 1+1=2 in cardinal numbers; fun fact, nobody has proved this in real numbers till date.) But deductive proofs are more fallible than empirical proofs Yet, you can't use empirical proofs in formal math. Haha. 2) Another thing is that formal math assumes two valued logic for deduction. But logic is not universal - it varies from culture to culture. (For example, Buddhists used a quasi truth functional logic called chatushkoti.) When you use a different logic than two valued logic, formal math fails. While I understand point (1) well, I will be honest in telling that I don't fully understand point (2) - quasi truth functional logic. I'm just regurgitating what I read in "the cultural foundations of mathematics" by CK Raju. I'll link one of his popular articles for you: www.boloji.com/articles/52924/california-indian-calculus
Oh, so I'm saying that formal math is inferior because it doesn't allow empirical proofs (which was allowed in many other ancient cultures). Formal math (axiomatic, mathematics as proof) is inferior to the normal math ( mathematics as calculation) which was widespread across various cultures such as Maya, Mesopotamia, India, etc. because the former doesn't allow empirical proofs while the latter does.
@@shinto2822 Thanks for clarifying that... you would probably enjoy reading some of Doron Zeilberger opinions. He is more or less in line with your point of view... and indeed, there are many other people who agree with you... thanks for sharing!
Thank you so much for your criticism! Historically, Fourier series were one of the main reasons for developing a solid foundation for calculus, and indeed they are a fantastic topic, and they could perfectly well fit an essay on the motivations for studying analysis! It is a great idea to do a video like the one you are suggesting! I might do a video on the historical reasons to develop analysis one day, and if I do so, I'd be sure to talk more about fourier series... This series of videos were done as a review for a more or less standard series of lectures on introduction to analysis and, for better of worst, most introduction to analysis courses tend to start with the axiomatic method instead of following the historical route... I find merits in both approaches though. As for the philosophy part, i feel that philosophy is often neglected by most mathematicians and students of mathematics... Most people focus on the technical parts, so I wanted to do something to help fill that gap. Of course, it does not to be for everyone, and i don't claim that my perspective is better than other people perspectives... In any case, I hope that you had found some value watching this video (and perhaps other videos as well), but if you didn't, take solace in the fact that the exploration process sometimes leads to dead ends... I do appreciate your comment as it gives me some ideas for a video on the historical development of analysis, and perhaps some in fourier analysis (it would be cool to get up to fourier analysis over groups and abstract harmonic analysis!). So Thank you!
BTW, I got to read your original comment as well, and you are absolutely right... At the very least, limit interchange should have been part of this video... That was definitely an omission on my part... A great book with that type of motivation, is the Analysis book by Tao... I hope most people will find such motivation as well... And if i re-make the video at some point, I'll be sure to include more of the historical perspective and those motivating problems. Thanks again for pointing that out!
I've been thinking hard about K-12 math in the USA and why it is such a train wreck/dumpster fire with worse and worse test scores. I've decided the main problem is because kids are never shown any sort of axiomatic, formalistic basis for what they're doing. Rather, they are hit with a big vertical wall of abstraction and never told what it is all about. They are taught math just as circus animals are taught circus animal trick, i.e., they are conditioned in a stimulus-response way. It's "When you see this, do this" math, and eventually they completely burn out. Circus animals never really understand what they're doing, only that punishment comes for not performing and reward comes from performing. Humans can't learn that way.
Great point! Your view is similar to the one that I have expressed in other videos... A now classic essay with a similar point of view is "A mathematician's lament" by Paul Lockhart. Paraphrasing Lockhart, if we had set up to make a system to kill the joy of learning, we couldn't have done a much better work than what we do right now.... But this is just the tip of the iceberg. For a while, the education system didn't make any sense to me until I realized that "education system" is a misnomer. It is a system system in which the goal is not to educate free thinkers but to produce future workers. But I am already sounding like an old man shaking my fist to a cloud and probably also wearing a tin foil hat, so I'll leave it at that for now!
@@academyofuselessideas Thanks for the kind words. I'll take them wherever I can get them. I surely haven't gotten any from any sort of education institution I've ever tried to work with. If you ever saw the film _Stand and Deliver_ there's a scene where the admin bureaucrat says to the math teacher "You'll crush what little self-esteem they have left" at his suggestion to teach underprivileged kids calculus. Right. And I'm coming from a CS angle. For incoming freshman CS majors they might have seen programming, and of course the whole math curriculum K-12, but they've never seen discrete math, which is usually a hodge-podge collection of higher, post-calc math topics -- and CS departments suffer very high dropout rates. For example, induction becomes recursion in programming. Where does K-12 deal with induction? Thousands more examples of circus trick style education failing. All in all math is very weak in the USA. If we trained our health professionals like we did our mathematicians there would be 1/1000th the doctors and nurses with sub-50% success rates. Just saying...
@@vonBottorff It is a very deep issue... I think that math shouldn't be mandatory at all... or for that matters, no class should be mandatory... instead teachers should strive to make courses entertaining enough for kids so they want to go without being forced... but all those are just dreams. A more realistic perspective might be to work within the system and help people who want to learn. Luckily, that's becoming more and more possible, there are so much pretty good material online which is accessible to anyone... if only their love for the subject does not get subdued, one can hope that they'll find the resources themselves... In any case, best luck in your journey, we all can make the world we want to see in the world!
Thanks for your encouraging words! Most of the videos are made for people who already took analysis, so they are a bit fast paced... but I am glad that you got something out of it!
😊 No reason to be sorry! Luigi is my third favorite character on the Mario Bros Franchise!... One of the main challenges I faced making videos is precisely my voice. Not being a native english speaker, and not being anywhere close to be a decent voice actor, I feel very limited with my voice range... I do my best and I enjoy the process... my hope is that my voice is a little more welcoming in newer videos though! I hope you find value in this one and other videos in the channel. Sometimes it hard to get an honest opinion so I thank you for your honesty!
Thanks for your feedback. Voice acting is one of the millions of things I don't know... but I want to learn! I'll do my best to improve my voice rhythm! Let us know if you see any improvement in more recent videos!
your handler reminds me of the great late Norm MacDonald... he had a joke that could go by that handler... Anyways, I am glad you find some use in the ideas talked about! and I am glad that you are not even mad despite feeling cheated!
0:15 Nuh uh, God made mat, we just discovered it. Proof? I feel like it. Also it makes math sound cooler cus u discovering shit rather than just hallucinating ideas outta nothing lol
very popular philosophical stand! having a philosophical stand towards math helps one to connect with the subject, so I am glad that you have one! I did a video on some philosophical views of math for those interested!
Well of course math exists in it’s entirety without us doing anything. How else would the laws of the universe actually work. However, our language of math, how we express it and how we communicate through it is a spice of the human endeavor. Our syntax is special to us. And I’m sure there’s quite a few different ways to write the math we know, some better, some worse.
@@corbinwilson660 Thank you so much for your take on this... it seems like you are some sort of Platonist who considers mathematics real but the language we use to express mathematics relative to our culture... That seems a solid position but one can go deeper... It is possible that the laws of physics are the real thing, and mathematics just happens to be good to express those laws of physics (The distinction is subtle here). We designed a huge part of math to express the laws of physics so we expect that math would be good for that... but there are large parts of math that don't seem well aligned with reality. An easy example is the Banach-Tarski paradox which says that you can cut a sphere in 8 parts, rearrange those parts and get two spheres of the same size than the original. The physical world don't seem to work like that at all (or at least, we haven't observed that), so is the Banach-Tarski paradox real?... another way of thinking about this (which is basically the same way but from a different perspective) is which math is real? there are different versions of mathematics, for example in geometry, you can do euclidean and non-euclidean geometry. For a long time, we thought that the world was euclidean but now we believe that it is not... so which one of the two mathematics (if any) is real? or are the two of them real in the world of ideas? Those are the types of questions that one could ask in the philosophy of mathematics... I thought i would share those just for fun
Bro the name of your channel is not helping you. I know it’s suppose to be satirical, but it just makes people think “ok if it’s useless I’ll go watch something else instead”.
This is a pretty good point which bring us to the question of the utility of math... I prefer to evaluate math by its intellectual challenge than by its utility. Similarly of how chess, despite being useless to plan real battles, can be pretty entertaining.
I don't really think that. If people want to learn useful information they often wouldn't watch random informational videos on completely random things. "Useful" information would usually be random life hacks and videos on things that school students might often need
Feel free to ask any questions or to share your opinions! I am considering remaking this video, and your feedback will be invaluable!
This is not a question. It is a demand. I am planning to self study Real Analysis in 2024. Please don't call it pointless; because it has. In fact, it has many.
@@AutoDisheep I am glad that you find real analysis useful! and in the end, isn't that what matters? because useful and useless are fairly subjective. I have little use for a fishing pole but in the hands of an experienced fisher-person it is a fantastic tool! Not only that, but usefulness also depends on your goals. If my goal is to learn russian, i have no use for a french-english dictionary. I hope that the videos help you in your journey (i wanted to make a whole introduction to real analysis series but I got distracted). As you advance, you'll notice the ironic nature of the "uselessness" claims... However, one of the main points is that there are things worth pursuing out of aesthetic value instead out of usefulness (at least as people usually define usefulness)...
Anyways, I wish you the best in your real analysis journey... if you have any real analysis questions (or any other question) feel free to join our discord server... one of the main issues with self studying real analysis (or advance math) is that most books don't give much of a motivation for why the topics are relevant, or interesting. It is much easier when you have people to talk about the things you are learning... I can also provide you with a list of recommended books and topics depending on your reasons to study the subject!
@@academyofuselessideas I already have several analysis book that I will dive into. However, if you would be kind enough to recommend me the best analysis book, and tell me why you like it.
@@AutoDisheep It would be easier if you tell me which books you have, so I can tell you about their strengths and weaknesses, and even discuss what problems are good from each one of them. However, I wrote a more detailed answer to your question as a post in the community section of the channel. As I was writing my answer, I realized that it was a bit too long for a youtube comment. Moreover, as a community post, some other people in the channel would be able to expand and give their own perspective on what is the best way to learn analysis. Please check that post out (or let me know if you cannot find it!)
It’s good to note analysis does have direct applications (albeit the maths is advanced) in things like image processing
True, some theorems in analysis have found interesting applications (even some theorems that go beyond calculus)... it might be nice to make a video on theorems that turned out to be applied to some real life problems! feel free to share some examples!
Grateful for having stumbled upon your channel. Pretty cool stuff!
It is my pleasure to talk about these subjects! Feel free to suggest any other topic, or to comment on your own interests!
Perfect introduction. If I had you as a teacher, I'd had learned Analysis years ago!
Thanks for your encouraging words... Most of my real analysis videos are fairly fast passed since they were originally intended as quick reviews more than for learning the topics for the first time... I might do some more detailed videos in analysis at some point though... in any case, if you have any question about analysis, or any type of math, we would love to talk about it in our discord server... i hope that your previous professors didn't discourage you from learning math or analysis in particular... if they did, forgive them as they didn't know what they were doing (and it is likely that they had the best intentions)... Thanks again for the encouraging words!
i had him as a teacher, and indeed, I learned a lot! Buena, Pedro!! Percibo muchas mejoras con la entonación!! Siga adelante con ewtos buenos contenidos!
@@seebaastian Gracias por el alentador comentario. Es interesante saber que de algo sirvieron mis clases! Tambien me alegra que haya encontrado el canal Mucha suerte con todo y un cordial saludo!
Me lo sugirió una muy buena amiga en común ;)
@@seebaastian
cool video!
I still mostly care about applied, but
a) I'm not comfortable with just accepting the tools as they are without fully understanding them, or
b) I'm not sure how I can ever have any confidence in any derived result without having confidence in the starting axioms, and for some domains, something like the results of calculus are the starting axioms...
Confidence may be the wrong word, I'm sure other people have checked them and that they work, I just don't know how I can build something that means something to me, out of things I might not be able to unambiguously interpret. That said, I am happy sometimes with a hand-wavy/intuitive notion (at least to start with, though it wouldn't hurt to get better at formalising and checking these).
I also like how you describe an almost engineering/computer security approach to reducing dependence on axioms here, as in reducing the amount of moving parts/points of failure, or the attack surface, after all, we want our maths to be secure/robust to logical attack.
yes! some people are comfortable using some results and intuitions, but others prefer to have a very solid foundation... in terms of the turtles analogy, some people are fine standing on the first turtle while others want to go deeper and deeper until they find what is standing in the last turtle... going deep down is quite challenging though, even for something like analysis, it is not that trivial to find the correct axioms... even some people who teach analysis don't go all the way down (and usually in a first analysis class no one goes all the way down because it would be too hard and overwhelming).... But at the very least, we can go down one turtle at a time
My opinion:
Well for sure, we have to accept logic, otherwise no statement can be made and it is impossible to anylize the world. We cannot even deny logic without using it.
After that, for axioms in other mathematical fields, we create them so that they can be a reliable way to describe some concept found in physical world, then using logic we can derive other scenarios and be 100% sure that they are as much reliable. (Which is where i seek confidence)
For more abstract parts of mathematics, we use the same laws that we already have and play with them further, so that they are still logically correct even if they no longer have basis in the real world.
@@MilkywayWarrior1618 pretty solid... we can even question logic, or using different logics to produce some other types of math... for example, you can deny the law of excluded middle (why? there are good reasons for denying that axiom... ask more if interested!) and work in intuitionistic logic... many of the results will agree with those of classical logic but you will need to work harder to find proofs.
but in general, i think that math started as an attempt of explaining physical phenomena, and that someway some how, humans realize that those mechanisms of reasoning applied as well to abstract objects.
Thanks for sharing!
@@academyofuselessideas I am interested, how can one question logic?
@@MilkywayWarrior1618 Pretty tough question that gets to a very deep part of the philosophy of mathematics. The example I gave is probably one of the most common challenges to classical logic (classical logic is just the name of the regular logic that you are probably aware of with propositions that you can form from some basic proposition and by using conjunctions (and), disjunctions (or), negation (not). in classic logic, the idea is that you can determine the truth value of each proposition (telling if a proposition is truth or false), and if you know the truth value of every basic proposition, you can find the truth value of using the "rules of logic"... One of those rules of (classical) logic is the rule of excluded middle, which says, that for every proposition p, you have that p or not p must be truth. In intuitionistic logic, that statement is not necessarily truth. The reasons for that are philosophical. A way of thinking about it is that in intuitionistic logic, propositions with truth values are propositions for which you can construct a proof. The fact that you cannot construct a proof for p does not mean that you can construct a proof for p... So intuitionists reject the law of excluded middle which makes their proofs often more complicated than the proofs done in classical logic... I am oversimplifying here a lot for the sake of the explanation, and there are things that i am not being 100% accurate here just to get the general idea across... You can find more on intuitionistic logic on mathoverflow... or if you want to talk about it more with me, feel free to join our discord server... This gets very tricky, and my short explanation here might not be the best one but I hope it shed some light, or at the very least, I hope that it is though provoking!
Wow, amazing video!
Thank you!
The very first part of Euclid Book I is a list of 23 definitions.
Then 5 axioms and 5 'common notions', which really act like axioms.
Book II adds two more definitions and so on. I am not sure why you think the axiomatic method doesn't use definitions - as it truly does.
Also - there are several equivalent definitions of the real numbers, for example, as Dedekind Cuts.
Pretty great observation... I should have going deeper into what i meant. If the goal is to provide foundations to calculus (or to any theory) you can take two different approaches. One is to provide consistent axioms that describe the theory (for lack of a better term, let's call this the axiomatic approach). The second approach is to reduce your theory to a simpler theory that is more "solid" (again, for lack of a better term, let's call this the constructive approach)... This approaches are not mutually exclusive but they are more like a rough sketch on how to approach theories. Mathematicians believe that all mathematical theories can be reduced to (or constructed from) set theory. Calculus is not different. You can reduce everything we say here to a conversation about sets. But set theory itself is describe in terms of axioms (usually Zermelo-Fraenkel axiomatic system, but there are other options... and there are indeed other options for a foundation than set theory but such discussion would take us too far). The approach here is axiomatic (as opposite to constructive) in the sense that instead of creating the natural numbers from sets, the integers from the naturals, the rationals from the integers, and finally the reals from the rationals (via Dedekind cuts or equivalence classes of Cauchy sequences (which by the way, we made another short video sketching that construction)), we start directly with axioms for the real numbers (namely, we start with the axioms for a complete ordered field)...
Of course, even if we have done all those constructions, one could ask "how do you construct sets?"... Here the analogy of the infinite regression of turtles applies again... The existence of sets is a very deep question in the philosophy of mathematics (the so called question of the ontology of sets). It is pretty interesting but something that most practitioners don't care too much about. Indeed, most practitioners don't even care to reduce math to set theory. They are happy knowing that such reduction is possible in principle.
When I argue that we don't "define" the real numbers, I should have been a bit more explicit, and perhaps a better wording is to say that we do not "construct the real numbers" from simpler mathematical objects... When I say that the axiomatic method does not define objects, perhaps a better wording would have been that "the axiomatic method does not tell you what an object is but it tells you how it behaves"... This is a more modern view than the one in Euclid's elements. Euclid probably thought that he was indeed defining points and lines... but eventually mathematicians realized that his definition didn't really say what those objects were, it only said what they do, and gave a possible interpretation for that object. Probably the most influential mathematician to reevaluate the work of Euclid was Hilbert. Indeed, most modern treatments of Euclid's Elements are based on Hilbert's version (this is because in perspective, Euclid's original work was not as mature as how we humans understand math nowadays).
All this points are very subtle, and most likely I am not explaining them that well... So, perhaps, an example might help... The first definition on Euclid's elements reads: "A point is that which has no part"... Does that really say what a point is? perhaps, but as you read more and more into Euclid's elements, you kind of realize that such definition does not play much of a role. It definitely serves to help the reader have an interpretation of what a point is, the interpretation helps to guide the other axioms, but if you accept the axioms, and the rules of logic, you will accept the theorems that Euclid proves even if you don't interpret what a point is in that way... indeed, you can find other interpretations of "point" that have nothing to do with the "standard" interpretation... it is kind of crazy, but that was Hilbert's realization (he was one of the first one to give a different interpretation of point and line!)...
Anyways, this comment is super long and probably very confusing... But if it piques your interest and you'd like to talk more about it, feel free to join our discord server! I'd love to hear more about your opinion since you raise a very interesting point... Moreover, I am in my own learning path, and the best way to improve one knowledge (in my opinion) is by talking more about this ideas!... Thank you so much for your insightful comment and for kindling the thoughts that I am sharing in this answer!
Very good channel
thanks for the encouraging words!
Cool
You don't need real numbers for notions of continuity and limits. You need a topology.
@IsomerSoma Thanks for sharing that... you might enjoy this series video on topology and continuity... Indeed, topology is the preferred way to look at continuity however coming up with the concept of continuity without any examples in mind is pretty hard. I don't even know if I could come up with the idea of topology on my own... but we have the benefit of our previous generations studying this problem and distilling the idea of a topology for us... Mathematical concepts come from human minds, and I find it fascinating how we figure out some of these abstract ideas!
Very cool video! Would it be possible for you to create a similar video about why study linear algebra?
Definitely! The motivations for linear algebra tend to be "easier" in a way... if you want an answer soon, feel free to join our discord and ask there, I'll give you my answer... or join one of our streams and ask... it might not be as polished as a video but I should be able to give you my perspective on the spot!
@@academyofuselessideas Thanks! I just like the way you present the topic in the video. By the way, I'm self-studying linear algebra, and I'm just curious about any intuition or history or how it evolved that kind of stuff other then the proofs in textbooks.
@@user-ih2ue4bw5q It's nice that you are selfstudying linear algebra. What materials are you using for that? If you are on discord, we can discuss this more... many books don't give enough motivation and you can easily get lost on the abstractions. Linear algebra is a great topic because it is formal but you have tons of examples and applications, so with good materials, you shouldn't have troubles staying in track... I can't think of any textbook based on a historical perspective, but if I come up with one I'll let you know... I'll also ask over discord to see if someone else in the community has a suggestion... I might also ask it in the community tab of the channel!
Enjoyed the vid.
Two things:
1) Why dont you start with the ZFC Axioms instead of the Peano Axioms? ZFC are the Axioms in a ton of other math branches. Therefore analysis results can be used in other branches, because they are built on the same axioms. (and vice versa)
2) Imo you missed one important advantage of axioms. (and why a solid foundation is absolutely necessary)
Rigour
Everyone is agreeing on the axioms and is therefore starting with the same rules. theorems can be checked! Axioms provide as much rigour as possible. Imagine everyone is just starting with different assumptions. Also one should keep in mind why rigour is so important: Logic tells us, that a wrong statement can imply any other statement. (Trump is female => im 3m tall) is a true statement. Thats kinda scary if you consider how long those implications-chains go starting by the axioms. One Mistake and everything can be concluded.
Do I really wanna sit in a plane which is based on flawed math in which everything can be concluded? Hell na
thank you so much for sharing! great points.
I'll take your suggestions into account if I make a longer version of the video... and I might even go further on a discussion of what's the best foundation for all the things we do in analysis... Originally, I thought of doing something like that, or do a series called "Analysis done wrong" which would start with subsystems of second order arithmetic and use only the axioms that you really need to prove some important analysis results (in the style of the "reverse mathematics" program)... But I realized that such style is mostly of interest of people deeply concerned with foundations (and that's why i wanted to call the series "Analysis done wrong" because it wouldn't be a good series for beginners).
There are some issues starting with ZFC... the main problem is if you go that deep, then you probably need to start with first order logic, and that would be a lot of work if the intention is to learn analysis... That might be why building the naturals from sets is often left for a course in Set theory than for a course in Analysis. But it would've worth to mention that all modern mathematics share ZF as it's foundation... as its ultimate turtle...
Your second point is pretty great... obviously rigor is a pretty good argument but you also mention something that to me seems as important and worth mentioning... having a common ground and language... that is, mathematicians know what kind of things are acceptable as part of a proof and an argument... Great point.
Thanks for taking the time to share your opinion... it is pretty insightful and thought provoking
Real analysis lays the foundation for calculus. But it is called ADVANCED CALCULUS.
I never thought about it but that's a little funny. It reminds me of a Bertrand Russel observation. He notices that studying math is similar to seeing things. Humans have a natural range of vision for which they don't need much effort seeing things, but if they want to see very large things far away they need telescopes, and if they want to see very small things they need microscopes. So Russel says that math is alike, there is a level of math that people can naturally address (in this example calculus), but if you want to see far you need to develop more and more tools (like differential equations, functional analysis, etc), and if you want to see smaller things you need also to develop tools (like real analysis). I guess that's why it is called advanced calculus, because it is harder to see than calculus even though it lays its foundations, but that's just my 2 cents. Your observation is very funny and insightful!
I cant belive terapagos terastilized math already ono
🤔
@@academyofuselessideas turtles all the way down
I never really understood calculus until I read a book on nonstandard analysis which introduces hyperreal numbers. For an engineer, it is much more intuitive. I really can't understand why schools still start with limits to build up calculus when hyperreal numbers are so much more intuitive. And as a bonus, it has been proven as mathematically rigorous as real analysis. Guess teachers are stuck in their ways just like most of us.
cool perspective... using infinitesimals is definitely a great way to think about calculus! Thanks for sharing this perspective!
The axiomatic method of proofs is inferior mathematics, just makes math more difficult to understand without adding any practical value to it. You said it yourself in the beginning of this video, real world applications don't require real numbers nor axioms.
Very interesting take... It's pretty hard to think of any modern mathematics that is not based or inspired or influenced by the axiomatic method. Which makes me realize that I don't understand your meaning of the word "mathematics", which in turn makes me realize that I don't understand your meaning of the word "inferior". To make sense of inferior, you need a criteria: A fish is inferior at flying than a bird, but a bird is inferior at swimming than a fish.
I do agree that, depending on your goals, the axiomatic method makes mathematics more difficult to understand. And I do think that if you don't see its value, then most likely you are better off not worrying about it. I would say that part of the problem is that the axiomatic method is so widespread in modern mathematics that mathematicians often don't even bother explaining why and under which circumstances the axiomatic method is necessary. Unfortunately, that is a mistake that I have made myself here. This gives me an idea for a video on "why mathematicians bother with the axiomatic method" but a short answer to that question is that mathematicians don't have any other alternative since deep down, whatever idea you construct comes from some beliefs (whether you are conscious of those beliefs or not is a different question). Mathematicians make their beliefs explicit in the form of axioms.
Pretty interesting and insightful and thought-provoking comment... I have some other thoughts but I need to learn to be more succinct in my comments!
@@academyofuselessideas
1) Formal math is based on the belief that deductive proofs are infallible. This is why you start by assuming a system of axioms and then deduce theorems from those axioms.
But, deductive proofs are not in reality infallible - those axioms could be wrong (which means that your theorems deduced from those axioms are wrong). Formal math fails because humans make errors in deduction (example - the best chess players often make mistakes in deduction). Empirical proofs are not allowed in formal math (which is why it took hundreds of pages for burtrend Russel to prove 1+1=2 in cardinal numbers; fun fact, nobody has proved this in real numbers till date.) But deductive proofs are more fallible than empirical proofs Yet, you can't use empirical proofs in formal math. Haha.
2) Another thing is that formal math assumes two valued logic for deduction. But logic is not universal - it varies from culture to culture. (For example, Buddhists used a quasi truth functional logic called chatushkoti.) When you use a different logic than two valued logic, formal math fails.
While I understand point (1) well, I will be honest in telling that I don't fully understand point (2) - quasi truth functional logic. I'm just regurgitating what I read in "the cultural foundations of mathematics" by CK Raju.
I'll link one of his popular articles for you:
www.boloji.com/articles/52924/california-indian-calculus
Oh, so I'm saying that formal math is inferior because it doesn't allow empirical proofs (which was allowed in many other ancient cultures).
Formal math (axiomatic, mathematics as proof) is inferior to the normal math ( mathematics as calculation) which was widespread across various cultures such as Maya, Mesopotamia, India, etc. because the former doesn't allow empirical proofs while the latter does.
@@shinto2822 Thanks for clarifying that... you would probably enjoy reading some of Doron Zeilberger opinions. He is more or less in line with your point of view... and indeed, there are many other people who agree with you... thanks for sharing!
@@academyofuselessideas thanks for the suggestion! I'll do as you have suggested.
I think the video would be more interesting if you talked more about fourier series and less about axiomatic systems and also less about philosophy
Thank you so much for your criticism! Historically, Fourier series were one of the main reasons for developing a solid foundation for calculus, and indeed they are a fantastic topic, and they could perfectly well fit an essay on the motivations for studying analysis! It is a great idea to do a video like the one you are suggesting! I might do a video on the historical reasons to develop analysis one day, and if I do so, I'd be sure to talk more about fourier series...
This series of videos were done as a review for a more or less standard series of lectures on introduction to analysis and, for better of worst, most introduction to analysis courses tend to start with the axiomatic method instead of following the historical route... I find merits in both approaches though.
As for the philosophy part, i feel that philosophy is often neglected by most mathematicians and students of mathematics... Most people focus on the technical parts, so I wanted to do something to help fill that gap. Of course, it does not to be for everyone, and i don't claim that my perspective is better than other people perspectives... In any case, I hope that you had found some value watching this video (and perhaps other videos as well), but if you didn't, take solace in the fact that the exploration process sometimes leads to dead ends...
I do appreciate your comment as it gives me some ideas for a video on the historical development of analysis, and perhaps some in fourier analysis (it would be cool to get up to fourier analysis over groups and abstract harmonic analysis!). So Thank you!
BTW, I got to read your original comment as well, and you are absolutely right... At the very least, limit interchange should have been part of this video... That was definitely an omission on my part... A great book with that type of motivation, is the Analysis book by Tao... I hope most people will find such motivation as well... And if i re-make the video at some point, I'll be sure to include more of the historical perspective and those motivating problems. Thanks again for pointing that out!
@@academyofuselessideas No problem I'll keep an eye out for your videos. I'd be a consumer of that type of content. Goodluck
I've been thinking hard about K-12 math in the USA and why it is such a train wreck/dumpster fire with worse and worse test scores. I've decided the main problem is because kids are never shown any sort of axiomatic, formalistic basis for what they're doing. Rather, they are hit with a big vertical wall of abstraction and never told what it is all about. They are taught math just as circus animals are taught circus animal trick, i.e., they are conditioned in a stimulus-response way. It's "When you see this, do this" math, and eventually they completely burn out. Circus animals never really understand what they're doing, only that punishment comes for not performing and reward comes from performing. Humans can't learn that way.
Great point!
Your view is similar to the one that I have expressed in other videos... A now classic essay with a similar point of view is "A mathematician's lament" by Paul Lockhart.
Paraphrasing Lockhart, if we had set up to make a system to kill the joy of learning, we couldn't have done a much better work than what we do right now.... But this is just the tip of the iceberg. For a while, the education system didn't make any sense to me until I realized that "education system" is a misnomer. It is a system system in which the goal is not to educate free thinkers but to produce future workers.
But I am already sounding like an old man shaking my fist to a cloud and probably also wearing a tin foil hat, so I'll leave it at that for now!
@@academyofuselessideas Thanks for the kind words. I'll take them wherever I can get them. I surely haven't gotten any from any sort of education institution I've ever tried to work with. If you ever saw the film _Stand and Deliver_ there's a scene where the admin bureaucrat says to the math teacher "You'll crush what little self-esteem they have left" at his suggestion to teach underprivileged kids calculus. Right. And I'm coming from a CS angle. For incoming freshman CS majors they might have seen programming, and of course the whole math curriculum K-12, but they've never seen discrete math, which is usually a hodge-podge collection of higher, post-calc math topics -- and CS departments suffer very high dropout rates. For example, induction becomes recursion in programming. Where does K-12 deal with induction? Thousands more examples of circus trick style education failing. All in all math is very weak in the USA. If we trained our health professionals like we did our mathematicians there would be 1/1000th the doctors and nurses with sub-50% success rates. Just saying...
@@vonBottorff It is a very deep issue... I think that math shouldn't be mandatory at all... or for that matters, no class should be mandatory... instead teachers should strive to make courses entertaining enough for kids so they want to go without being forced... but all those are just dreams. A more realistic perspective might be to work within the system and help people who want to learn. Luckily, that's becoming more and more possible, there are so much pretty good material online which is accessible to anyone... if only their love for the subject does not get subdued, one can hope that they'll find the resources themselves... In any case, best luck in your journey, we all can make the world we want to see in the world!
i learned more in this 4 minutes than two semesters of studying real analysis lol. good video
Thanks for your encouraging words! Most of the videos are made for people who already took analysis, so they are a bit fast paced... but I am glad that you got something out of it!
The video is awesome but I am so sorry I have to say you sound a bit like luigi 😂
😊 No reason to be sorry! Luigi is my third favorite character on the Mario Bros Franchise!... One of the main challenges I faced making videos is precisely my voice. Not being a native english speaker, and not being anywhere close to be a decent voice actor, I feel very limited with my voice range... I do my best and I enjoy the process... my hope is that my voice is a little more welcoming in newer videos though! I hope you find value in this one and other videos in the channel. Sometimes it hard to get an honest opinion so I thank you for your honesty!
You talk very fast. Great video though.
Thanks for your feedback. Voice acting is one of the millions of things I don't know... but I want to learn! I'll do my best to improve my voice rhythm! Let us know if you see any improvement in more recent videos!
Channel is called Academy of Useless Ideas
But the ideas are not useless
I feel cheated but I'm not even mad
your handler reminds me of the great late Norm MacDonald... he had a joke that could go by that handler... Anyways, I am glad you find some use in the ideas talked about! and I am glad that you are not even mad despite feeling cheated!
🤣
Glad you liked it! that's the appropriate response 🙃
0:15
Nuh uh, God made mat, we just discovered it.
Proof? I feel like it. Also it makes math sound cooler cus u discovering shit rather than just hallucinating ideas outta nothing lol
very popular philosophical stand! having a philosophical stand towards math helps one to connect with the subject, so I am glad that you have one! I did a video on some philosophical views of math for those interested!
underage moment
Well of course math exists in it’s entirety without us doing anything. How else would the laws of the universe actually work. However, our language of math, how we express it and how we communicate through it is a spice of the human endeavor. Our syntax is special to us. And I’m sure there’s quite a few different ways to write the math we know, some better, some worse.
@@corbinwilson660 Thank you so much for your take on this... it seems like you are some sort of Platonist who considers mathematics real but the language we use to express mathematics relative to our culture... That seems a solid position but one can go deeper... It is possible that the laws of physics are the real thing, and mathematics just happens to be good to express those laws of physics (The distinction is subtle here).
We designed a huge part of math to express the laws of physics so we expect that math would be good for that... but there are large parts of math that don't seem well aligned with reality. An easy example is the Banach-Tarski paradox which says that you can cut a sphere in 8 parts, rearrange those parts and get two spheres of the same size than the original. The physical world don't seem to work like that at all (or at least, we haven't observed that), so is the Banach-Tarski paradox real?... another way of thinking about this (which is basically the same way but from a different perspective) is which math is real? there are different versions of mathematics, for example in geometry, you can do euclidean and non-euclidean geometry. For a long time, we thought that the world was euclidean but now we believe that it is not... so which one of the two mathematics (if any) is real? or are the two of them real in the world of ideas?
Those are the types of questions that one could ask in the philosophy of mathematics... I thought i would share those just for fun
I mean this viewpoint sounds naive today but it drove many of the greatest mathematicians (cauchy and newton immediately spring to mind)
Bro the name of your channel is not helping you. I know it’s suppose to be satirical, but it just makes people think “ok if it’s useless I’ll go watch something else instead”.
This is a pretty good point which bring us to the question of the utility of math... I prefer to evaluate math by its intellectual challenge than by its utility. Similarly of how chess, despite being useless to plan real battles, can be pretty entertaining.
Litterally nobody thinks that.
Its useless because its useless. its still nice to learn tho.
I don't really think that. If people want to learn useful information they often wouldn't watch random informational videos on completely random things. "Useful" information would usually be random life hacks and videos on things that school students might often need
Your comment is very stup1d.
@@academyofuselessideas the name of you channel is just great.